Landmark point selection

ABSTRACT

An exemplary method comprises receiving data points, selecting a first subset of the data points to generate an initial set of landmarks, each data point of the first subset defining a landmark point and for each non-landmark data point: calculating first data point distances between a respective non-landmark data point and each landmark point of the initial set of landmarks, identifying a first shortest data point distance from among the first data point distances between the respective non-landmark data point and each landmark point of the initial set of landmarks, and storing the first shortest data point distance as a first landmark distance for the respective non-landmark data point. The method further comprising identifying a non-landmark data point with a longest first landmark distance in comparison with other first landmark distances and adding the identified non-landmark data point associated as a first landmark point to the initial set of landmarks.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part and claims priority to U.S.patent application Ser. No. 14/639,954, filed Mar. 5, 2015, entitled“Systems and Methods for Capture of Relationships Within Information,”which claims priority to U.S. Provisional Patent Application Ser. No.61/948,490, filed Mar. 5, 2014, entitled “Systems and Methods forCapture of Relationships Within Information,” and are incorporated byreference herein.

BACKGROUND

1. Field of the Invention(s)

Embodiments discussed herein are directed to grouping of data points fordata analysis and more particularly to generating a graph utilizingimproved groupings of data points based on scores of the groupings.

2. RELATED ART

As the collection and storage data has increased, there is an increasedneed to analyze and make sense of large amounts of data. Examples oflarge datasets may be found in financial services companies, oilexpiration, biotech, and academia. Unfortunately, previous methods ofanalysis of large multidimensional datasets tend to be insufficient (ifpossible at all) to identify important relationships and may becomputationally inefficient.

In order to process large datasets, some previous methods of analysisuse clustering. Clustering often breaks important relationships and isoften too blunt an instrument to assist in the identification ofimportant relationships in the data. Similarly, previous methods oflinear regression, projection pursuit, principal component analysis, andmultidimensional scaling often do not reveal important relationships.Further, existing linear algebraic and analytic methods are toosensitive to large scale distances and, as a result, lose detail.

Even if the data is analyzed, sophisticated experts are often necessaryto interpret and understand the output of previous methods. Althoughsome previous methods allow graphs that depict some relationships in thedata, the graphs are not interactive and require considerable time for ateam of such experts to understand the relationships. Further, theoutput of previous methods does not allow for exploratory data analysiswhere the analysis can be quickly modified to discover newrelationships. Rather, previous methods require the formulation of ahypothesis before testing.

SUMMARY OF THE INVENTION(S)

An example method comprises receiving data points, selecting a firstsubset of the data points to generate an initial set of landmarks, eachdata point of the first subset defining a landmark point, for eachnon-landmark data point: calculating first data point distances betweena respective non-landmark data point and each landmark point of theinitial set of landmarks, identifying a first shortest data pointdistance from among the first data point distances between therespective non-landmark data point and each landmark point of theinitial set of landmarks, and storing the first shortest data pointdistance as a first landmark distance for the respective non-landmarkdata point, identifying a non-landmark data point with a longest firstlandmark distance in comparison with other first landmark distances ofother non-landmark data points, and adding the non-landmark data pointassociated with the longest first landmark distance as a first landmarkpoint to the initial set of landmarks to generate an expanded set oflandmark points.

The method may further comprise storing a set of longest landmarkdistances for the non-landmark data points of the expanded set oflandmark points, calculating second data point distances between thenon-landmark data points associated with the set of longest landmarkdistances and each landmark point of the expanded set of landmarkpoints, the expanded set of landmark points including the first landmarkpoint, and adding the non-landmark data point associated with thelongest second landmark distance to the expanded set of landmark points,thereby categorizing the non-landmark data point as a new landmarkpoint. In various embodiments, the method may further comprisedetermining if the expanded set of landmark points includes apredetermined number of landmark points, if the expanded set of landmarkpoints includes less than the predetermined number of landmark points:storing another set of longest second landmark distances for thenon-landmark data points of the expanded set of landmark points,calculating another data point distances between the non-landmark datapoints associated with the set of longest second landmark distances andeach landmark point of the first and second sets of landmark points, thefirst set and second set of landmark points including the first andsecond landmark point, and adding the non-landmark data point associatedwith the longest second landmark distance as a another landmark point tothe set of landmark points, and if the expanded set of landmark pointsincludes less than the predetermined number of landmark points, generatea visualization of nodes and edges using the expanded set of landmarkpoints instead of the received data points.

The expanded set of landmark points may be an approximation of thereceived data points. The method may comprise, before the first subsetof the data points are selected to generate an initial set of landmarks:accessing a data structure containing information, generating amathematical reference space, and mapping a similarity space using theinformation from the data structure into the mathematical referencespace to generate the data points, wherein each data point is defined bythe mapping into the reference space. In some embodiments, the methodmay further comprise generating a cover for the landmark points of theexpanded set of landmark points in the mathematical reference spacebased on a resolution, clustering the landmark points of the expandedset of landmark points into subsets using a metric to generate subsetsof the landmark points of the expanded set of landmark points todetermine each individual node of a plurality of nodes, each of thenodes of the plurality of nodes comprising members representative of atleast one subset of the landmark points of the expanded set of landmarkpoints, and generating an interactive visualization comprising nodes anda plurality of edges wherein each of the edges of the plurality of edgesconnects nodes with shared members.

In some embodiments, the method may further comprise generating a coverfor the data points in the mathematical reference space based on aresolution, clustering the data points into subsets using a metric togenerate subsets of the data points to determine each individual node ofa plurality of nodes, each of the nodes of the plurality of nodescomprising members representative of at least one subset of the datapoints, each node defining a data point, and generating an interactivevisualization comprising nodes of landmark points of the expanded set oflandmark points and a plurality of edges wherein each of the edges ofthe plurality of edges connects nodes with shared members.

The selection of the first subset of the data point may be random.Receiving data points may comprise storing the data points in a memorysystem and the initial set of landmarks is stored in a non-transitorystorage system. Identifying the non-landmark data point with the longestfirst landmark distance in comparison with other first landmarkdistances of other non-landmark data points may comprise identifying apredetermined number of non-landmark data points, each of thepredetermined number having longer first landmark distances incomparison with the other first landmark distances of the othernon-landmark data points. Adding the non-landmark data point associatedwith the longest first landmark distance as a first landmark point tothe initial set of landmarks may comprise adding the predeterminednumber of non-landmark data points associated with the longer firstlandmark distances to the initial set of landmarks.

An example non-transitory computer readable medium may compriseinstructions executable by a processor to perform a method. The methodmay comprise: receiving data points, selecting a first subset of thedata points to generate an initial set of landmarks, each data point ofthe first subset defining a landmark point, for each non-landmark datapoint: calculating first data point distances between a respectivenon-landmark data point and each landmark point of the initial set oflandmarks, identifying a first shortest data point distance from amongthe first data point distances between the respective non-landmark datapoint and each landmark point of the initial set of landmarks, andstoring the first shortest data point distance as a first landmarkdistance for the respective non-landmark data point, identifying anon-landmark data point with a longest first landmark distance incomparison with other first landmark distances of other non-landmarkdata points, and adding the non-landmark data point associated with thelongest first landmark distance as a first landmark point to the initialset of landmarks to generate an expanded set of landmark points.

An example system comprises an input module, a random landmark selectionmodule, a distance calculation module, a landmark distance comparisonmodule, and a landmark assignment module. The input module may beconfigured to receive data points. The random landmark selection modulemay be configured to select a first subset of the data points togenerate an initial set of landmarks, each data point of the firstsubset defining a landmark point. The distance calculation module may beconfigured to, for each non-landmark data point: calculate first datapoint distances between a respective non-landmark data point and eachlandmark point of the initial set of landmarks, identify a firstshortest data point distance from among the first data point distancesbetween the respective non-landmark data point and each landmark pointof the initial set of landmarks, and store the first shortest data pointdistance as a first landmark distance for the respective non-landmarkdata point. The landmark distance comparison module may be configured toidentify a non-landmark data point with a longest first landmarkdistance in comparison with other first landmark distances of othernon-landmark data points. The landmark assignment module may beconfigured to add the non-landmark data point associated with thelongest first landmark distance as a first landmark point to the initialset of landmarks to generate an expanded set of landmark points.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is an example graph representing data that appears to be dividedinto three disconnected groups.

FIG. 1B is an example graph representing data set obtained from aLotka-Volterra equation modeling the populations of predators and preyover time.

FIG. 1C is an example graph of data sets whereby the data does not breakup into disconnected groups, but instead has a structure in which thereare lines (or flares) emanating from a central group.

FIG. 2 is an example environment in which embodiments may be practiced.

FIG. 3 is a block diagram of an example analysis server.

FIG. 4 is a flow chart depicting an example method of dataset analysisand visualization in some embodiments.

FIG. 5 is an example ID field selection interface window in someembodiments.

FIG. 6 a is an example data field selection interface window in someembodiments.

FIG. 6 b is an example metric and filter selection interface window insome embodiments.

FIG. 7 is an example filter parameter interface window in someembodiments.

FIG. 8 is a flowchart for data analysis and generating a visualizationin some embodiments.

FIG. 9 is an example interactive visualization in some embodiments.

FIG. 10 is an example interactive visualization displaying an explaininformation window in some embodiments.

FIG. 11 is a flowchart of functionality of the interactive visualizationin some embodiments.

FIG. 12 is a flowchart of for generating a cancer map visualizationutilizing biological data of a plurality of patients in someembodiments.

FIG. 13 is an example data structure including biological data for anumber of patients that may be used to generate the cancer mapvisualization in some embodiments.

FIG. 14 is an example visualization displaying the cancer map in someembodiments.

FIG. 15 is a flowchart of for positioning new patient data relative tothe cancer map visualization in some embodiments.

FIG. 16 is an example visualization displaying the cancer map includingpositions for three new cancer patients in some embodiments.

FIG. 17 is a flowchart of utilization the visualization and positioningof new patient data in some embodiments

FIG. 18 is an example digital device in some embodiments.

FIG. 19 shows an example landmark module configured to identify landmarkpoints that approximate or represent a larger collection of data pointsin accordance with various embodiments.

FIG. 20 is a flowchart for generating a set of landmark points in someembodiments.

FIG. 21A shows example metric space containing data in accordance withvarious embodiments.

FIG. 21B shows subset composed of individual data points in accordancewith some embodiments.

FIG. 21C shows example random landmarks R₁, R₂, R₃, and R₄ that havebeen randomly selected as initial landmarks in the subset identified inFIG. 21A.

FIG. 21D shows lines corresponding to data point distances to eachlandmark for three points (P₁, P₂, and P₃) in the subset identified inFIG. 21A.

FIG. 22A shows example data point distances between point P₁ and randomlandmarks R₁, R₂, R₃, and R₄.

FIG. 22B shows example distances between point P₂ and random landmarksR₁, R₂, R₃, and R₄.

FIG. 22C shows an example table wherein distances for each point arestored.

FIG. 23A shows example landmark distances for points P₁, P₂, and P₃ tolandmark R₁ which can be used to demonstrate the selection of additionallandmark points.

FIG. 23B shows example shortest distances from each non-landmark pointto each landmark point.

FIG. 23C shows point P₂ as new MM landmark point L₁ in this example.

FIG. 23D shows subset with L₁ as a new landmark where the distancesbetween various points have been calculated.

FIG. 24A shows an example wherein data in X does not fit into localmemory (e.g., Random Access Memory (RAM)) and is, therefore, read off oflong term storage.

FIG. 24B shows an example wherein data point sets are stored in localmemory instead of the landmark set in accordance with variousembodiments.

FIG. 25A shows subset with distances shown for points P₁, P₂, P₃, and P₄to their respective closest random landmark (R₁, R₂, R₃, R₄).

FIG. 25B shows example shortest distances from each non-landmark pointto each landmark point.

FIG. 25C shows points P₂, P₃, and P₄ as landmarks L₁, L₂, and L₃ in thisexample

DETAILED DESCRIPTION OF DRAWINGS

Some embodiments described herein may be a part of the subject ofTopological Data Analysis (TDA). TDA is an area of research which hasproduced methods for studying point cloud data sets from a geometricpoint of view. Other data analysis techniques use “approximation bymodels” of various types. Examples of other data analysis techniquesinclude regression methods which model data as a graph of a function inone or more variables. Unfortunately, certain qualitative properties(which one can readily observe when the data is two-dimensional) may beof a great deal of importance for understanding, and these features maynot be readily represented within such models.

FIG. 1A is an example graph representing data that appears to be dividedinto three disconnected groups. In this example, the data for this graphmay be associated with various physical characteristics related todifferent population groups or biomedical data related to differentforms of a disease. Seeing that the data breaks into groups in thisfashion can give insight into the data, once one understands whatcharacterizes the groups.

FIG. 1B is an example graph representing data set obtained from aLotka-Volterra equation modeling the populations of predators and preyover time. From FIG. 1B, one observation about this data is that it isarranged in a loop. The loop is not exactly circular, but it istopologically a circle. The exact form of the equations, whileinteresting, may not be of as much importance as this qualitativeobservation which reflects the fact that the underlying phenomenon isrecurrent or periodic. When looking for periodic or recurrent phenomena,methods may be developed which can detect the presence of loops withoutdefining explicit models. For example, periodicity may be detectablewithout having to first develop a fully accurate model of the dynamics.

FIG. 1C is an example graph of data sets whereby the data does not breakup into disconnected groups, but instead has a structure in which thereare lines (or flares) emanating from a central group. In this case, thedata also suggests the presence of three distinct groups, but theconnectedness of the data does not reflect this. This particular datathat is the basis for the example graph in FIG. 1C arises from a studyof single nucleotide polymorphisms (SNPs).

In each of the examples above, aspects of the shape of the data arerelevant in reflecting information about the data. Connectedness (thesimplest property of shape) reflects the presence of a discreteclassification of the data into disparate groups. The presence of loops,another simple aspect of shape, often reflect periodic or recurrentbehavior. Finally, in the third example, the shape containing flaressuggests a classification of the data descriptive of ways in whichphenomena can deviate from the norm, which would typically berepresented by the central core. These examples support the idea thatthe shape of data (suitably defined) is an important aspect of itsstructure, and that it is therefore important to develop methods foranalyzing and understanding its shape. The part of mathematics whichconcerns itself with the study of shape is called topology, andtopological data analysis attempts to adapt methods for studying shapewhich have been developed in pure mathematics to the study of the shapeof data, suitably defined.

One question is how notions of geometry or shape are translated intoinformation about point clouds, which are, after all, finite sets? Whatwe mean by shape or geometry can come from a dissimilarity function ormetric (e.g., a non-negative, symmetric, real-valued function d on theset of pairs of points in the data set which may also satisfy thetriangle inequality, and d(x; y)=0 if and only if x=y). Such functionsexist in profusion for many data sets. For example, when data comes inthe form of a numerical matrix, where the rows correspond to the datapoints and the columns are the fields describing the data, then-dimensional Euclidean distance function is natural when there are nfields. Similarly, in this example, there are Pearson correlationdistances, cosine distances, and other choices.

When the data is not Euclidean, for example if one is consideringgenomic sequences, various notions of distance may be defined usingmeasures of similarity based on Basic Local Alignment Search Tool(BLAST) type similarity scores. Further, a measure of similarity cancome in non-numeric forms, such as social networks of friends orsimilarities of hobbies, buying patterns, tweeting, and/or professionalinterests. In any of these ways the notion of shape may be formulatedvia the establishment of a useful notion of similarity of data points.

One of the advantages of TDA is that TDA may depend on nothing more thansuch a notion, which is a very primitive or low-level model. TDA mayrely on many fewer assumptions than standard linear or algebraic models,for example. Further, the methodology may provide new ways ofvisualizing and compressing data sets, which facilitate understandingand monitoring data. The methodology may enable study ofinterrelationships among disparate data sets and/ormultiscale/multiresolution study of data sets. Moreover, the methodologymay enable interactivity in the analysis of data, using point and clickmethods.

In some embodiments, TDA may be a very useful complement to moretraditional methods, such as Principal Component Analysis (PCA),multidimensional scaling, and hierarchical clustering. These existingmethods are often quite useful, but suffer from significant limitations.PCA, for example, is an essentially linear procedure and there aretherefore limits to its utility in highly non-linear situations.Multidimensional scaling is a method which is not intrinsically linear,but can in many situations wash out detail, since it may overweightlarge distances. In addition, when metrics do not satisfy an intrinsicflatness condition, it may have difficulty in faithfully representingthe data. Hierarchical clustering does exhibit multiscale behavior, butrepresents data only as disjoint clusters, rather than retaining any ofthe geometry of the data set. In all four cases, these limitationsmatter for many varied kinds of data.

We now summarize example properties of an example construction, in someembodiments, which may be used for representing the shape of data setsin a useful, understandable fashion as a finite graph:

-   -   The input may be a collection of data points equipped in some        way with a distance or dissimilarity function, or other        description. This can be given implicitly when the data is in        the form of a matrix, or explicitly as a matrix of distances or        even the generating edges of a mathematical network.    -   One construction may also use one or more lens functions (i.e.        real valued functions on the data). Lens function(s) may depend        directly on the metric. For example, lens function(s) might be        the result of a density estimator or a measure of centrality or        data depth. Lens function(s) may, in some embodiments, depend on        a particular representation of the data, as when one uses the        first one or two coordinates of a principal component or        multidimensional scaling analysis. In some embodiments, the lens        function(s) may be columns which expert knowledge identifies as        being intrinsically interesting, as in cholesterol levels and        BMI in a study of heart disease.    -   In some embodiments, the construction may depend on a choice of        two or more processing parameters, resolution, and gain.        Increase in resolution typically results in more nodes and an        increase in the gain increases the number of edges in a        visualization and/or graph in a reference space as further        described herein.    -   The output may be, for example, a visualization (e.g., a display        of connected nodes or “network”) or simplicial complex. One        specific combinatorial formulation in one embodiment may be that        the vertices form a finite set, and then the additional        structure may be a collection of edges (unordered pairs of        vertices) which are pictured as connections in this network.

In various embodiments, a system for handling, analyzing, andvisualizing data using drag and drop methods as opposed to text basedmethods is described herein. Philosophically, data analytic tools arenot necessarily regarded as “solvers,” but rather as tools forinteracting with data. For example, data analysis may consist of severaliterations of a process in which computational tools point to regions ofinterest in a data set. The data set may then be examined by people withdomain expertise concerning the data, and the data set may then besubjected to further computational analysis. In some embodiments,methods described herein provide for going back and forth betweenmathematical constructs, including interactive visualizations (e.g.,graphs), on the one hand and data on the other.

In one example of data analysis in some embodiments described herein, anexemplary clustering tool is discussed which may be more powerful thanexisting technology, in that one can find structure within clusters andstudy how clusters change over a period of time or over a change ofscale or resolution.

An example interactive visualization tool (e.g., a visualization modulewhich is further described herein) may produce combinatorial output inthe form of a graph which can be readily visualized. In someembodiments, the example interactive visualization tool may be lesssensitive to changes in notions of distance than current methods, suchas multidimensional scaling.

Some embodiments described herein permit manipulation of the data from avisualization. For example, portions of the data which are deemed to beinteresting from the visualization can be selected and converted intodatabase objects, which can then be further analyzed. Some embodimentsdescribed herein permit the location of data points of interest withinthe visualization, so that the connection between a given visualizationand the information the visualization represents may be readilyunderstood.

FIG. 2 is an example environment 200 in which embodiments may bepracticed. In various embodiments, data analysis and interactivevisualization may be performed locally (e.g., with software and/orhardware on a local digital device), across a network (e.g., via cloudcomputing), or a combination of both. In many of these embodiments, adata structure is accessed to obtain the data for the analysis, theanalysis is performed based on properties and parameters selected by auser, and an interactive visualization is generated and displayed. Thereare many advantages between performing all or some activities locallyand many advantages of performing all or some activities over a network.

Environment 200 comprises user devices 202 a-202 n, a communicationnetwork 204, data storage server 206, and analysis server 208.Environment 200 depicts an embodiment wherein functions are performedacross a network. In this example, the user(s) may take advantage ofcloud computing by storing data in a data storage server 206 over acommunication network 204. The analysis server 208 may perform analysisand generation of an interactive visualization.

User devices 202 a-202 n may be any digital devices. A digital device isany device that includes memory and a processor. Digital devices arefurther described in FIG. 18. The user devices 202 a-202 n may be anykind of digital device that may be used to access, analyze and/or viewdata including, but not limited to a desktop computer, laptop, notebook,or other computing device.

In various embodiments, a user, such as a data analyst, may generateand/or receive a database or other data structure with the user device202 a to be saved to the data storage server 206. The user device 202 amay communicate with the analysis server 208 via the communicationnetwork 204 to perform analysis, examination, and visualization of datawithin the database.

The user device 202 a may comprise any number of client programs. One ormore of the client programs may interact with one or more applicationson the analysis server 208. In other embodiments, the user device 202 amay communicate with the analysis server 208 using a browser or otherstandard program. In various embodiments, the user device 202 acommunicates with the analysis server 208 via a virtual private network.Those skilled in the art will appreciate that that communication betweenthe user device 202 a, the data storage server 206, and/or the analysisserver 208 may be encrypted or otherwise secured.

The communication network 204 may be any network that allows digitaldevices to communicate. The communication network 204 may be theInternet and/or include LAN and WANs. The communication network 204 maysupport wireless and/or wired communication.

The data storage server 206 is a digital device that is configured tostore data. In various embodiments, the data storage server 206 storesdatabases and/or other data structures. The data storage server 206 maybe a single server or a combination of servers. In one example the datastorage server 206 may be a secure server wherein a user may store dataover a secured connection (e.g., via https). The data may be encryptedand backed-up. In some embodiments, the data storage server 206 isoperated by a third-party such as Amazon's S3 service.

The database or other data structure may comprise large high-dimensionaldatasets. These datasets are traditionally very difficult to analyzeand, as a result, relationships within the data may not be identifiableusing previous methods. Further, previous methods may be computationallyinefficient.

The analysis server 208 may include any number of digital devicesconfigured to analyze data (e.g., the data in the stored database and/orother dataset received and/or generated by the user device 202 a).Although only one digital device is depicted in FIG. 2 corresponding tothe analysis server 208, it will be appreciated that any number offunctions of the analysis server 208 may be performed by any number ofdigital devices.

In various embodiments, the analysis server 208 may perform manyfunctions to interpret, examine, analyze, and display data and/orrelationships within data. In some embodiments, the analysis server 208performs, at least in part, topological analysis of large datasetsapplying metrics, filters, and resolution parameters chosen by the user.The analysis is further discussed regarding FIG. 8 herein.

The analysis server 208 may generate graphs in memory, visualizedgraphs, and/or an interactive visualization of the output of theanalysis. The interactive visualization allows the user to observe andexplore relationships in the data. In various embodiments, theinteractive visualization allows the user to select nodes comprisingdata that has been clustered. The user may then access the underlyingdata, perform further analysis (e.g., statistical analysis) on theunderlying data, and manually reorient the graph(s) (e.g., structures ofnodes and edges described herein) within the interactive visualization.The analysis server 208 may also allow for the user to interact with thedata, see the graphic result. The interactive visualization is furtherdiscussed in FIGS. 9-11.

The graphs in memory and/or visualized graphs may also include nodesand/or edges as described herein. Graphs that are generated in memorymay not be depicted to a user but rather may be in memory of a digitaldevice. Visualized graphs are rendered graphs that may be depicted tothe user (e.g., using user device 202 a).

In some embodiments, the analysis server 208 interacts with the userdevice(s) 202 a-202 n over a private and/or secure communicationnetwork. The user device 202 a may include a client program that allowsthe user to interact with the data storage server 206, the analysisserver 208, another user device (e.g., user device 202 n), a database,and/or an analysis application executed on the analysis server 208.

It will be appreciated that all or part of the data analysis may occurat the user device 202 a. Further, all or part of the interaction withthe visualization (e.g., graphic) may be performed on the user device202 a. Alternately, all or part of the data analysis may occur on anynumber of digital devices including, for example, on the analysis server208.

Although two user devices 202 a and 202 n are depicted, those skilled inthe art will appreciate that there may be any number of user devices inany location (e.g., remote from each other). Similarly, there may be anynumber of communication networks, data storage servers, and analysisservers.

Cloud computing may allow for greater access to large datasets (e.g.,via a commercial storage service) over a faster connection. Further,those skilled in the art will appreciate that services and computingresources offered to the user(s) may be scalable.

FIG. 3 is a block diagram of an example analysis server 208. In someembodiments, the analysis server 208 comprises a processor 302,input/output (I/O) interface 304, a communication network interface 306,a memory system 308, a storage system 310, and a processing module 312.The processor 302 may comprise any processor or combination ofprocessors with one or more cores.

The input/output (I/O) interface 304 may comprise interfaces for variousI/O devices such as, for example, a keyboard, mouse, and display device.The example communication network interface 306 is configured to allowthe analysis server 208 to communication with the communication network204 (see FIG. 2). The communication network interface 306 may supportcommunication over an Ethernet connection, a serial connection, aparallel connection, and/or an ATA connection. The communication networkinterface 306 may also support wireless communication (e.g., 802.11a/b/g/n, WiMax, LTE, WiFi). It will be apparent to those skilled in theart that the communication network interface 306 can support many wiredand wireless standards.

The memory system 308 may be any kind of memory including RAM, ROM, orflash, cache, virtual memory, etc. In various embodiments, working datais stored within the memory system 308. The data within the memorysystem 308 may be cleared or ultimately transferred to the storagesystem 310.

The storage system 310 includes any storage configured to retrieve andstore data. Some examples of the storage system 310 include flashdrives, hard drives, optical drives, and/or magnetic tape. Each of thememory system 308 and the storage system 310 comprises a non-transitorycomputer-readable medium, which stores instructions (e.g., softwareprograms) executable by processor 302.

The storage system 310 comprises a plurality of modules utilized byembodiments of discussed herein. A module may be hardware, software(e.g., including instructions executable by a processor), or acombination of both. In one embodiment, the storage system 310 includesa processing module 312. The processing module 312 may include memoryand/or hardware and includes an input module 314, a filter module 316, aresolution module 318, an analysis module 320, a visualization engine322, and database storage 324. Alternative embodiments of the analysisserver 208 and/or the storage system 310 may comprise more, less, orfunctionally equivalent components and modules.

The input module 314 may be configured to receive commands andpreferences from the user device 202 a. In various examples, the inputmodule 314 receives selections from the user which will be used toperform the analysis. The output of the analysis may be an interactivevisualization.

The input module 314 may provide the user a variety of interface windowsallowing the user to select and access a database, choose fieldsassociated with the database, choose a metric, choose one or morefilters, and identify resolution parameters for the analysis. In oneexample, the input module 314 receives a database identifier andaccesses a large multidimensional database. The input module 314 mayscan the database and provide the user with an interface window allowingthe user to identify an ID field. An ID field is an identifier for eachdata point. In one example, the identifier is unique. The same columnname may be present in the table from which filters are selected. Afterthe ID field is selected, the input module 314 may then provide the userwith another interface window to allow the user to choose one or moredata fields from a table of the database.

Although interactive windows may be described herein, those skilled inthe art will appreciate that any window, graphical user interface,and/or command line may be used to receive or prompt a user or userdevice 202 a for information.

The filter module 316 may subsequently provide the user with aninterface window to allow the user to select a metric to be used inanalysis of the data within the chosen data fields. The filter module316 may also allow the user to select and/or define one or more filters.

The resolution module 318 may allow the user to select a resolution,including filter parameters. In one example, the user enters a number ofintervals and a percentage overlap for a filter.

The analysis module 320 may perform data analysis based on the databaseand the information provided by the user. In various embodiments, theanalysis module 320 performs an algebraic topological analysis toidentify structures and relationships within data and clusters of data.Those skilled in the art will appreciate that the analysis module 320may use parallel algorithms or use generalizations of variousstatistical techniques (e.g., generalizing the bootstrap to zig-zagmethods) to increase the size of data sets that can be processed. Theanalysis is further discussed herein (e.g., see discussion regardingFIG. 8). It will be appreciated that the analysis module 320 is notlimited to algebraic topological analysis but may perform any analysis.

The visualization engine 322 generates an interactive visualizationbased on the output from the analysis module 320. The interactivevisualization allows the user to see all or part of the analysisgraphically. The interactive visualization also allows the user tointeract with the visualization. For example, the user may selectportions of a graph from within the visualization to see and/or interactwith the underlying data and/or underlying analysis. The user may thenchange the parameters of the analysis (e.g., change the metric,filter(s), or resolution(s)) which allows the user to visually identifyrelationships in the data that may be otherwise undetectable using priormeans. The interactive visualization is further described herein (e.g.,see discussion regarding FIGS. 9-11).

The database storage 324 is configured to store all or part of thedatabase that is being accessed. In some embodiments, the databasestorage 324 may store saved portions of the database. Further, thedatabase storage 324 may be used to store user preferences, parameters,and analysis output thereby allowing the user to perform many differentfunctions on the database without losing previous work.

Those skilled in the art will appreciate that that all or part of theprocessing module 312 may be at the user device 202 a or the databasestorage server 206. In some embodiments, all or some of thefunctionality of the processing module 312 may be performed by the userdevice 202 a.

In various embodiments, systems and methods discussed herein may beimplemented with one or more digital devices. In some examples, someembodiments discussed herein may be implemented by a computer program(instructions) executed by a processor. The computer program may providea graphical user interface. Although such a computer program isdiscussed, those skilled in the art will appreciate that embodiments maybe performed using any of the following, either alone or in combination,including, but not limited to, a computer program, multiple computerprograms, firmware, and/or hardware.

A module and/or engine may include any processor or combination ofprocessors. In some examples, a module and/or engine may include or be apart of a processor, digital signal processor (DSP), applicationspecific integrated circuit (ASIC), an integrated circuit, and/or thelike. In various embodiments, the module and/or engine may be softwareor firmware.

FIG. 4 is a flow chart 400 depicting an example method of datasetanalysis and visualization in some embodiments. In step 402, the inputmodule 314 accesses a database. The database may be any data structurecontaining data (e.g., a very large dataset of multidimensional data).In some embodiments, the database may be a relational database. In someexamples, the relational database may be used with MySQL, Oracle,Microsoft SQL Server, Aster nCluster, Teradata, and/or Vertica. Thoseskilled in the art will appreciate that the database may not be arelational database.

In some embodiments, the input module 314 receives a database identifierand a location of the database (e.g., the data storage server 206) fromthe user device 202 a (see FIG. 2). The input module 314 may then accessthe identified database. In various embodiments, the input module 314may read data from many different sources, including, but not limited toMS Excel files, text files (e.g., delimited or CSV), Matlab .mat format,or any other file.

In some embodiments, the input module 314 receives an IP address orhostname of a server hosting the database, a username, password, and thedatabase identifier. This information (herein referred to as “connectioninformation”) may be cached for later use. It will be appreciated thatthe database may be locally accessed and that all, some, or none of theconnection information may be required. In one example, the user device202 a may have full access to the database stored locally on the userdevice 202 a so the IP address is unnecessary. In another example, theuser device 202 a may already have loaded the database and the inputmodule 314 merely begins by accessing the loaded database.

In various embodiments, the identified database stores data withintables. A table may have a “column specification” which stores the namesof the columns and their data types. A “row” in a table, may be a tuplewith one entry for each column of the correct type. In one example, atable to store employee records might have a column specification suchas:

-   -   employee_id primary key int (this may store the employee's ID as        an integer, and uniquely identifies a row)    -   age int    -   gender char(1) (gender of the employee may be a single character        either M or F)    -   salary double (salary of an employee may be a floating point        number)    -   name varchar (name of the employee may be a variable-length        string)        In this example, each employee corresponds to a row in this        table. Further, the tables in this example relational database        are organized into logical units called databases. An analogy to        file systems is that databases can be thought of as folders and        files as tables. Access to databases may be controlled by the        database administrator by assigning a username/password pair to        authenticate users.

Once the database is accessed, the input module 314 may allow the userto access a previously stored analysis or to begin a new analysis. Ifthe user begins a new analysis, the input module 314 may provide theuser device 202 a with an interface window allowing the user to identifya table from within the database. In one example, the input module 314provides a list of available tables from the identified database.

In step 404, the input module 314 receives a table identifieridentifying a table from within the database. The input module 314 maythen provide the user with a list of available ID fields from the tableidentifier. In step 406, the input module 314 receives the ID fieldidentifier from the user and/or user device 202 a. The ID field is, insome embodiments, the primary key.

Having selected the primary key, the input module 314 may generate a newinterface window to allow the user to select data fields for analysis.In step 408, the input module 314 receives data field identifiers fromthe user device 202 a. The data within the data fields may be lateranalyzed by the analysis module 320.

In step 408, the filter module 316 selects one or more filters. In someembodiments, the filter module 316 and/or the input module 314 generatesan interface window allowing the user of the user device 202 a optionsfor a variety of different metrics and filter preferences. The interfacewindow may be a drop down menu identifying a variety of distance metricsto be used in the analysis.

In some embodiments, the user selects and/or provides filteridentifier(s) to the filter module 316. The role of the filters in theanalysis is also further described herein. The filters, for example, maybe user defined, geometric, or based on data which has beenpre-processed. In some embodiments, the data based filters are numericalarrays which can assign a set of real numbers to each row in the tableor each point in the data generally.

A variety of geometric filters may be available for the user to choose.Geometric filters may include, but are not limited to:

-   -   Density    -   L1 Eccentricity    -   L-infinity Eccentricity    -   Witness based Density    -   Witness based Eccentricity    -   Eccentricity as distance from a fixed point    -   Approximate Kurtosis of the Eccentricity

In step 410, the filter module 316 identifies a metric. Metric optionsmay include, but are not limited to, Euclidean, DB Metric, variancenormalized Euclidean, and total normalized Euclidean. The metric and theanalysis are further described herein.

In step 412, the resolution module 318 defines the resolution to be usedwith a filter in the analysis. The resolution may comprise a number ofintervals and an overlap parameter. In various embodiments, theresolution module 318 allows the user to adjust the number of intervalsand overlap parameter (e.g., percentage overlap) for one or morefilters.

In step 414, the analysis module 320 processes data of selected fieldsbased on the metric, filter(s), and resolution(s) to generate thevisualization. This process is further discussed herein (e.g., seediscussion regarding FIG. 8).

In step 416, the visualization engine 322 displays the interactivevisualization. In various embodiments, the visualization may be renderedin two or three dimensional space. The visualization engine 322 may usean optimization algorithm for an objective function which is correlatedwith good visualization (e.g., the energy of the embedding). Thevisualization may show a collection of nodes corresponding to each ofthe partial clusters in the analysis output and edges connecting them asspecified by the output. The interactive visualization is furtherdiscussed herein (e.g., see discussion regarding FIGS. 9-11).

Although many examples discuss the input module 314 as providinginterface windows, it will be appreciated that all or some of theinterface may be provided by a client on the user device 202 a. Further,in some embodiments, the user device 202 a may be running all or some ofthe processing module 312.

FIGS. 5-7 depict various interface windows to allow the user to makeselections, enter information (e.g., fields, metrics, and filters),provide parameters (e.g., resolution), and provide data (e.g., identifythe database) to be used with analysis. It will be appreciated that anygraphical user interface or command line may be used to make selections,enter information, provide parameters, and provide data.

FIG. 5 is an exemplary ID field selection interface window 500 in someembodiments. The ID field selection interface window 500 allows the userto identify an ID field. The ID field selection interface window 500comprises a table search field 502, a table list 504, and a fieldsselection window 506.

In various embodiments, the input module 314 identifies and accesses adatabase from the database storage 324, user device 202 a, or the datastorage server 206. The input module 314 may then generate the ID fieldselection interface window 500 and provide a list of available tables ofthe selected database in the table list 504. The user may click on atable or search for a table by entering a search query (e.g., a keyword)in the table search field 502. Once a table is identified (e.g., clickedon by the user), the fields selection window 506 may provide a list ofavailable fields in the selected table. The user may then choose a fieldfrom the fields selection window 506 to be the ID field. In someembodiments, any number of fields may be chosen to be the ID field(s).

FIG. 6 a is an example data field selection interface window 600 a insome embodiments. The data field selection interface window 600 a allowsthe user to identify data fields. The data field selection interfacewindow 600 a comprises a table search field 502, a table list 504, afields selection window 602, and a selected window 604.

In various embodiments, after selection of the ID field, the inputmodule 314 provides a list of available tables of the selected databasein the table list 504. The user may click on a table or search for atable by entering a search query (e.g., a keyword) in the table searchfield 502. Once a table is identified (e.g., clicked on by the user),the fields selection window 506 may provide a list of available fieldsin the selected table. The user may then choose any number of fieldsfrom the fields selection window 602 to be data fields. The selecteddata fields may appear in the selected window 604. The user may alsodeselect fields that appear in the selected window 604.

Those skilled in the art will appreciate that the table selected by theuser in the table list 504 may be the same table selected with regard toFIG. 5. In some embodiments, however, the user may select a differenttable. Further, the user may, in various embodiments, select fields froma variety of different tables.

FIG. 6 b is an example metric and filter selection interface window 600b in some embodiments. The metric and filter selection interface window600 b allows the user to identify a metric, add filter(s), and adjustfilter parameters. The metric and filter selection interface window 600b comprises a metric pull down menu 606, an add filter from databasebutton 608, and an add geometric filter button 610.

In various embodiments, the user may click on the metric pull down menu606 to view a variety of metric options. Various metric options aredescribed herein. In some embodiments, the user may define a metric. Theuser defined metric may then be used with the analysis.

In one example, finite metric space data may be constructed from a datarepository (i.e., database, spreadsheet, or Matlab file). This may meanselecting a collection of fields whose entries will specify the metricusing the standard Euclidean metric for these fields, when they arefloating point or integer variables. Other notions of distance, such asgraph distance between collections of points, may be supported.

The analysis module 320 may perform analysis using the metric as a partof a distance function. The distance function can be expressed by aformula, a distance matrix, or other routine which computes it. The usermay add a filter from a database by clicking on the add filter fromdatabase button 608. The metric space may arise from a relationaldatabase, a Matlab file, an Excel spreadsheet, or other methods forstoring and manipulating data. The metric and filter selection interfacewindow 600 b may allow the user to browse for other filters to use inthe analysis. The analysis and metric function are further describedherein (e.g., see discussion regarding FIG. 8).

The user may also add a geometric filter 610 by clicking on the addgeometric filter button 610. In various embodiments, the metric andfilter selection interface window 600 b may provide a list of geometricfilters from which the user may choose.

FIG. 7 is an example filter parameter interface window 700 in someembodiments. The filter parameter interface window 700 allows the userto determine a resolution for one or more selected filters (e.g.,filters selected in the metric and filter selection interface window600). The filter parameter interface window 700 comprises a filter namemenu 702, an interval field 704, an overlap bar 706, and a done button708.

The filter parameter interface window 700 allows the user to select afilter from the filter name menu 702. In some embodiments, the filtername menu 702 is a drop down box indicating all filters selected by theuser in the metric and filter selection interface window 600. Once afilter is chosen, the name of the filter may appear in the filter namemenu 702. The user may then change the intervals and overlap for one,some, or all selected filters.

The interval field 704 allows the user to define a number of intervalsfor the filter identified in the filter name menu 702. The user mayenter a number of intervals or scroll up or down to get to a desirednumber of intervals. Any number of intervals may be selected by theuser. The function of the intervals is further discussed herein (e.g.,see discussion regarding FIG. 8).

The overlap bar 706 allows the user to define the degree of overlap ofthe intervals for the filter identified in the filter name menu 702. Inone example, the overlap bar 706 includes a slider that allows the userto define the percentage overlap for the interval to be used with theidentified filter. Any percentage overlap may be set by the user.

Once the intervals and overlap are defined for the desired filters, theuser may click the done button. The user may then go back to the metricand filter selection interface window 600 and see a new option to runthe analysis. In some embodiments, the option to run the analysis may beavailable in the filter parameter interface window 700. Once theanalysis is complete, the result may appear in an interactivevisualization further described herein (e.g., see discussion regardingFIGS. 9-11).

It will be appreciated that interface windows in FIGS. 4-7 are examples.The example interface windows are not limited to the functional objects(e.g., buttons, pull down menus, scroll fields, and search fields)shown. Any number of different functional objects may be used. Further,as described herein, any other interface, command line, or graphicaluser interface may be used.

FIG. 8 is a flowchart 800 for data analysis and generating aninteractive visualization in some embodiments. In various embodiments,the processing on data and user-specified options is motivated bytechniques from topology and, in some embodiments, algebraic topology.These techniques may be robust and general. In one example, thesetechniques apply to almost any kind of data for which some qualitativeidea of “closeness” or “similarity” exists. The techniques discussedherein may be robust because the results may be relatively insensitiveto noise in the data and even to errors in the specific details of thequalitative measure of similarity, which, in some embodiments, may begenerally refer to as “the distance function” or “metric.” It will beappreciated that while the description of the algorithms below may seemgeneral, the implementation of techniques described herein may apply toany level of generality.

In step 802, the input module 314 receives data S. In one example, auser identifies a data structure and then identifies ID and data fields.Data S may be based on the information within the ID and data fields. Invarious embodiments, data S is treated as being processed as a finite“similarity space,” where data S has a real-valued function d defined onpairs of points s and t in S, such that:

d(s, s)=0

d(s, t)=d(t, s)

d(s, t)>=0

These conditions may be similar to requirements for a finite metricspace, but the conditions may be weaker. In various examples, thefunction is a metric.

It will be appreciated that data S may be a finite metric space, or ageneralization thereof, such as a graph or weighted graph. In someembodiments, data S be specified by a formula, an algorithm, or by adistance matrix which specifies explicitly every pairwise distance.

In step 804, the input module 314 generates reference space R. In oneexample, reference space R may be a well-known metric space (e.g., suchas the real line). The reference space R may be defined by the user. Instep 806, the analysis module 320 generates a map ref( ) from S into R.The map ref( ) from S into R may be called the “reference map.”

In one example, a reference of map from S is to a reference metric spaceR. R may be Euclidean space of some dimension, but it may also be thecircle, torus, a tree, or other metric space. The map can be describedby one or more filters (i.e., real valued functions on S). These filterscan be defined by geometric invariants, such as the output of a densityestimator, a notion of data depth, or functions specified by the originof S as arising from a data set.

In step 808, the resolution module 318 generates a cover of R based onthe resolution received from the user (e.g., filter(s), intervals, andoverlap—see discussion regarding FIG. 7 for example). The cover of R maybe a finite collection of open sets (in the metric of R) such that everypoint in R lies in at least one of these sets. In various examples, R isk-dimensional Euclidean space, where k is the number of filterfunctions. More precisely in this example, R is a box in k-dimensionalEuclidean space given by the product of the intervals [min_k, max_k],where min_k is the minimum value of the k-th filter function on S, andmax_k is the maximum value.

For example, suppose there are 2 filter functions, F1 and F2, and thatF1's values range from −1 to +1, and F2's values range from 0 to 5. Thenthe reference space is the rectangle in the x/y plane with corners(−1,0), (1,0), (−1, 5), (1, 5), as every point s of S will give rise toa pair (F1(s), F2(s)) that lies within that rectangle.

In various embodiments, the cover of R is given by taking products ofintervals of the covers of [min_k,max_k] for each of the k filters. Inone example, if the user requests 2 intervals and a 50% overlap for F1,the cover of the interval [−1,+1] will be the two intervals (−1.5, 0.5),(−0.5, 1.5). If the user requests 5 intervals and a 30% overlap for F2,then that cover of [0, 5] will be (−0.3, 1.3), (0.7, 2.3), (1.7, 3.3),(2.7, 4.3), (3.7, 5.3). These intervals may give rise to a cover of the2-dimensional box by taking all possible pairs of intervals where thefirst of the pair is chosen from the cover for F1 and the second fromthe cover for F2. This may give rise to 2*5, or 10, open boxes thatcovered the 2-dimensional reference space. However, those skilled in theart will appreciate that the intervals may not be uniform, or that thecovers of a k-dimensional box may not be constructed by products ofintervals. In some embodiments, there are many other choices ofintervals. Further, in various embodiments, a wide range of coversand/or more general reference spaces may be used.

In one example, given a cover, C₁, . . . , C_(m), of R, the referencemap is used to assign a set of indices to each point in S, which are theindices of the C_(j) such that ref(s) belongs to C_(j). This functionmay be called ref_tags(s). In a language such as Java, ref_tags would bea method that returned an int[ ]. Since the C's cover R in this example,ref(s) must lie in at least one of them, but the elements of the coverusually overlap one another, which means that points that “land near theedges” may well reside in multiple cover sets. In considering the twofilter example, if F1(s) is −0.99, and F2(s) is 0.001, then ref(s) is(−0.99, 0.001), and this lies in the cover element (−1.5,0.5)×(−0.3,1.3). Supposing that was labeled C₁, the reference map mayassign s to the set {1}. On the other hand, if t is mapped by F1, F2 to(0.1, 2.1), then ref(t) will be in (−1.5,0.5)×(0.7, 2.3), (−0.5,1.5)×(0.7,2.3), (−1.5,0.5)×(1.7,3.3), and (−0.5, 1.5)×(1.7,3.3), so theset of indices would have four elements for t.

Having computed, for each point, which “cover tags” it is assigned to,for each cover element, C_(d), the points may be constructed, whose tagsincluded, as set S(d). This may mean that every point s is in S(d) forsome d, but some points may belong to more than one such set. In someembodiments, there is, however, no requirement that each S(d) isnon-empty, and it is frequently the case that some of these sets areempty. In the non-parallelized version of some embodiments, each point xis processed in turn, and x is inserted into a hash-bucket for each j inref_tags(t) (that is, this may be how S(d) sets are computed).

It will be appreciated that the cover of the reference space R may becontrolled by the number of intervals and the overlap identified in theresolution (e.g., see further discussion regarding FIG. 7). For example,the more intervals, the finer the resolution in S—that is, the fewerpoints in each S(d), but the more similar (with respect to the filters)these points may be. The greater the overlap, the more times thatclusters in S(d) may intersect clusters in S(e)—this means that more“relationships” between points may appear, but, in some embodiments, thegreater the overlap, the more likely that accidental relationships mayappear.

In step 810, the analysis module 320 clusters each S(d) based on themetric, filter, and the space S. In some embodiments, a dynamicsingle-linkage clustering algorithm may be used to partition S(d). Itwill be appreciated that any number of clustering algorithms may be usedwith embodiments discussed herein. For example, the clustering schememay be k-means clustering for some k, single linkage clustering, averagelinkage clustering, or any method specified by the user.

The significance of the user-specified inputs may now be seen. In someembodiments, a filter may amount to a “forced stretching” in a certaindirection. In some embodiments, the analysis module 320 may not clustertwo points unless ALL of the filter values are sufficiently “related”(recall that while normally related may mean “close,” the cover mayimpose a much more general relationship on the filter values, such asrelating two points s and t if ref(s) and ref(t) are sufficiently closeto the same circle in the plane). In various embodiments, the ability ofa user to impose one or more “critical measures” makes this techniquemore powerful than regular clustering, and the fact that these filterscan be anything, is what makes it so general.

The output may be a simplicial complex, from which one can extract its1-skeleton. The nodes of the complex may be partial clusters, (i.e.,clusters constructed from subsets of S specified as the preimages ofsets in the given covering of the reference space R).

In step 812, the visualization engine 322 identifies nodes which areassociated with a subset of the partition elements of all of the S(d)for generating an interactive visualization. For example, suppose thatS={1, 2, 3, 4}, and the cover is C₁, C₂, C₃. Then if ref_tags(1)={1, 2,3} and ref_tags(2)={2, 3}, and ref_tags(3)={3}, and finallyref_tags(4)={1, 3}, then S(1) in this example is {1, 4}, S(2)={1,2}, andS(3)={1,2,3,4}. If 1 and 2 are close enough to be clustered, and 3 and 4are, but nothing else, then the clustering for S(1) may be {1} {3}, andfor S(2) it may be {1,2}, and for S(3) it may be {1,2}, {3,4}. So thegenerated graph has, in this example, at most four nodes, given by thesets {1}, {4}, {1,2}, and {3,4} (note that {1,2} appears in twodifferent clusterings). Of the sets of points that are used, two nodesintersect provided that the associated node sets have a non-emptyintersection (although this could easily be modified to allow users torequire that the intersection is “large enough” either in absolute orrelative terms).

Nodes may be eliminated for any number of reasons. For example, a nodemay be eliminated as having too few points and/or not being connected toanything else. In some embodiments, the criteria for the elimination ofnodes (if any) may be under user control or have application-specificrequirements imposed on it. For example, if the points are consumers,for instance, clusters with too few people in area codes served by acompany could be eliminated. If a cluster was found with “enough”customers, however, this might indicate that expansion into area codesof the other consumers in the cluster could be warranted.

In step 814, the visualization engine 322 joins clusters to identifyedges (e.g., connecting lines between nodes). Once the nodes areconstructed, the intersections (e.g., edges) may be computed “all atonce,” by computing, for each point, the set of node sets (not ref_tags,this time). That is, for each s in S, node_id_set(s) may be computed,which is an int[ ]. In some embodiments, if the cover is well behaved,then this operation is linear in the size of the set S, and we theniterate over each pair in node_id_set(s). There may be an edge betweentwo node_id's if they both belong to the same node_id_set( ) value, andthe number of points in the intersection is precisely the number ofdifferent node_id sets in which that pair is seen. This means that,except for the clustering step (which is often quadratic in the size ofthe sets S(d), but whose size may be controlled by the choice of cover),all of the other steps in the graph construction algorithm may be linearin the size of S, and may be computed quite efficiently.

In step 816, the visualization engine 322 generates the interactivevisualization of interconnected nodes (e.g., nodes and edges displayedin FIGS. 9 and 10).

It will be appreciated that it is possible, in some embodiments, to makesense in a fairly deep way of connections between various ref( ) mapsand/or choices of clustering. Further, in addition to computing edges(pairs of nodes), the embodiments described herein may be extended tocompute triples of nodes, etc. For example, the analysis module 320 maycompute simplicial complexes of any dimension (by a variety of rules) onnodes, and apply techniques from homology theory to the graphs to helpusers understand a structure in an automatic (or semi-automatic) way.

Further, it will be appreciated that uniform intervals in the coveringmay not always be a good choice. For example, if the points areexponentially distributed with respect to a given filter, uniformintervals can fail—in such case adaptive interval sizing may yielduniformly-sized S(d) sets, for instance.

Further, in various embodiments, an interface may be used to encodetechniques for incorporating third-party extensions to data access anddisplay techniques. Further, an interface may be used to for third-partyextensions to underlying infrastructure to allow for new methods forgenerating coverings, and defining new reference spaces.

FIG. 9 is an example interactive visualization 900 in some embodiments.The display of the interactive visualization may be considered a “graph”in the mathematical sense. The interactive visualization comprises oftwo types of objects: nodes (e.g., nodes 902 and 906) (which may beballs and may be colored) and the edges (e.g., edge 904) (the blacklines). The edges connect pairs of nodes (e.g., edge 904 connects node902 with node 906). As discussed herein, each node may represent acollection of data points (rows in the database identified by the user).In one example, connected nodes tend to include data points which are“similar to” (e.g., clustered with) each other. The collection of datapoints may be referred to as being “in the node.” The interactivevisualization may be two-dimensional, three-dimensional, or acombination of both.

In various embodiments, connected nodes and edges may form a graph orstructure. There may be multiple graphs in the interactivevisualization. In one example, the interactive visualization may displaytwo or more unconnected structures of nodes and edges.

The visual properties of the nodes and edges (such as, but not limitedto, color, stroke color, text, texture, shape, coordinates of the nodeson the screen) can encode any data based property of the data pointswithin each node. For example, coloring of the nodes and/or the edgesmay indicate (but is not limited to) the following:

-   -   Values of fields or filters    -   Any general functions of the data in the nodes (e.g., if the        data were unemployment rates by state, then GDP of the states        may be identifiable by color the nodes)    -   Number of data points in the node

The interactive visualization 900 may contain a “bar” 910 which maycomprise a legend indicating patterns and/or coloring of the nodes(e.g., balls) and may also identify what the patterns and/or colorsindicate. For example, in FIG. 9, bar 910 may indicate that color ofsome nodes is based on the density filter with blue (on the far left ofthe bar 910) indicating “4.99e+03” and red (on the far right of the bar910) indicating “1.43e+04.” In general this might be expanded to showany other legend by which nodes and/or edges are colored. It will beappreciated that, in some embodiments, the user may control the color aswell as what the color (and/or stroke color, text, texture, shape,coordinates of the nodes on the screen) indicates.

The user may also drag and drop objects of the interactive visualization900. In various embodiments, the user may reorient structures of nodesand edges by dragging one or more nodes to another portion of theinteractive visualization (e.g., a window). In one example, the user mayselect node 902, hold node 902, and drag the node across the window. Thenode 902 will follow the user's cursor, dragging the structure of edgesand/or nodes either directly or indirectly connected to the node 902. Insome embodiments, the interactive visualization 900 may depict multipleunconnected structures. Each structure may include nodes, however, noneof the nodes of either structure are connected to each other. If theuser selects and drags a node of the first structure, only the firststructure will be reoriented with respect to the user action. The otherstructure will remain unchanged. The user may wish to reorient thestructure in order to view nodes, select nodes, and/or better understandthe relationships of the underlying data.

In one example, a user may drag a node to reorient the interactivevisualization (e.g., reorient the structure of nodes and edges). Whilethe user selects and/or drags the node, the nodes of the structureassociated with the selected node may move apart from each other inorder to provide greater visibility. Once the user lets go (e.g.,deselects or drops the node that was dragged), the nodes of thestructure may continue to move apart from each other.

In various embodiments, once the visualization engine 322 generates theinteractive display, the depicted structures may move by spreading outthe nodes from each other. In one example, the nodes spread from eachother slowly allowing the user to view nodes distinguish from each otheras well as the edges. In some embodiments, the visualization engine 322optimizes the spread of the nodes for the user's view. In one example,the structure(s) stop moving once an optimal view has been reached.

It will be appreciated that the interactive visualization 900 mayrespond to gestures (e.g., multi-touch), stylus, or other interactionsallowing the user to reorient nodes and edges and/or interacting withthe underlying data.

The interactive visualization 900 may also respond to user actions suchas when the user drags, clicks, or hovers a mouse cursor over a node. Insome embodiments, when the user selects a node or edge, node informationor edge information may be displayed. In one example, when a node isselected (e.g., clicked on by a user with a mouse or a mouse cursorhovers over the node), a node information box 908 may appear thatindicates information regarding the selected node. In this example, thenode information box 908 indicates an ID, box ID, number of elements(e.g., data points associated with the node), and density of the dataassociated with the node.

The user may also select multiple nodes and/or edges by clickingseparate on each object, or drawing a shape (such as a box) around thedesired objects. Once the objects are selected, a selection informationbox 912 may display some information regarding the selection. Forexample, selection information box 912 indicates the number of nodesselected and the total points (e.g., data points or elements) of theselected nodes.

The interactive visualization 900 may also allow a user to furtherinteract with the display. Color option 914 allows the user to displaydifferent information based on color of the objects. Color option 914 inFIG. 9 is set to filter_Density, however, other filters may be chosenand the objects re-colored based on the selection. It will beappreciated that the objects may be colored based on any filter,property of data, or characterization. When a new option is chosen inthe color option 914, the information and/or colors depicted in thecolor bar 910 may be updated to reflect the change.

Layout checkbox 916 may allow the user to anchor the interactivevisualization 900. In one example, the layout checkbox 916 is checkedindicating that the interactive visualization 900 is anchored. As aresult, the user will not be able to select and drag the node and/orrelated structure. Although other functions may still be available, thelayout checkbox 916 may help the user keep from accidentally movingand/or reorienting nodes, edges, and/or related structures. It will beappreciated the layout checkbox 916 may indicate that the interactivevisualization 900 is anchored when the layout checkbox 916 is uncheckedand that when the layout checkbox 916 is checked the interactivevisualization 900 is no longer anchored.

The change parameters button 918 may allow a user to change theparameters (e.g., add/remove filters and/or change the resolution of oneor more filters). In one example, when the change parameters button 918is activated, the user may be directed back to the metric and filterselection interface window 600 (see FIG. 6) which allows the user to addor remove filters (or change the metric). The user may then view thefilter parameter interface 700 (see FIG. 7) and change parameters (e.g.,intervals and overlap) for one or more filters. The analysis module 320may then re-analyze the data based on the changes and display a newinteractive visualization 900 without again having to specify the datasets, filters, etc.

The find ID's button 920 may allow a user to search for data within theinteractive visualization 900. In one example, the user may click thefind ID's button 920 and receive a window allowing the user to identifydata or identify a range of data. Data may be identified by ID orsearching for the data based on properties of data and/or metadata. Ifdata is found and selected, the interactive visualization 900 mayhighlight the nodes associated with the selected data. For example,selecting a single row or collection of rows of a database orspreadsheet may produce a highlighting of nodes whose correspondingpartial cluster contains any element of that selection.

In various embodiments, the user may select one or more objects andclick on the explain button 922 to receive in-depth informationregarding the selection. In some embodiments, when the user selects theexplain button 922, the information about the data from which theselection is based may be displayed. The function of the explain button922 is further discussed herein (e.g., see discussion regarding FIG.10).

In various embodiments, the interactive visualization 900 may allow theuser to specify and identify subsets of interest, such as outputfiltering, to remove clusters or connections which are too small orotherwise uninteresting. Further, the interactive visualization 900 mayprovide more general coloring and display techniques, including, forexample, allowing a user to highlight nodes based on a user-specifiedpredicate, and coloring the nodes based on the intensity ofuser-specified weighting functions.

The interactive visualization 900 may comprise any number of menu items.The “Selection” menu may allow the following functions:

-   -   Select singletons (select nodes which are not connected to other        nodes)    -   Select all (selects all the nodes and edges)    -   Select all nodes (selects all nodes)    -   Select all edges    -   Clear selection (no selection)    -   Invert Selection (selects the complementary set of nodes or        edges)    -   Select “small” nodes (allows the user to threshold nodes based        on how many points they have)    -   Select leaves (selects all nodes which are connected to long        “chains” in the graph)    -   Remove selected nodes    -   Show in a table (shows the selected nodes and their associated        data in a table)    -   Save selected nodes (saves the selected data to whatever format        the user chooses. This may allow the user to subset the data and        create new data sources which may be used for further analysis.)

In one example of the “show in a table” option, information from aselection of nodes may be displayed. The information may be specific tothe origin of the data. In various embodiments, elements of a databasetable may be listed, however, other methods specified by the user mayalso be included. For example, in the case of microarray data from geneexpression data, heat maps may be used to view the results of theselections.

The interactive visualization 900 may comprise any number of menu items.The “Save” menu may allow may allow the user to save the whole output ina variety of different formats such as (but not limited to):

-   -   Image files (PNG/JPG/PDF/SVG etc.)    -   Binary output (The interactive output is saved in the binary        format. The user may reopen this file at any time to get this        interactive window again)

In some embodiments, graphs may be saved in a format such that thegraphs may be used for presentations. This may include simply saving theimage as a pdf or png file, but it may also mean saving an executable.xml file, which may permit other users to use the search and savecapability to the database on the file without having to recreate theanalysis.

In various embodiments, a relationship between a first and a secondanalysis output/interactive visualization for differing values of theinterval length and overlap percentage may be displayed. The formalrelationship between the first and second analysis output/interactivevisualization may be that when one cover refines the next, there is amap of simplicial complexes from the output of the first to the outputof the second. This can be displayed by applying a restricted form of athree-dimensional graph embedding algorithm, in which a graph is theunion of the graphs for the various parameter values and in which theconnections are the connections in the individual graphs as well asconnections from one node to its image in the following graph. Theconstituent graphs may be placed in its own plane in 3D space. In someembodiments, there is a restriction that each constituent graph remainwithin its associated plane. Each constituent graph may be displayedindividually, but a small change of parameter value may result in thevisualization of the adjacent constituent graph. In some embodiments,nodes in the initial graph will move to nodes in the next graph, in areadily visualizable way.

FIG. 10 is an example interactive visualization 1000 displaying anexplain information window 1002 in some embodiments. In variousembodiments, the user may select a plurality of nodes and click on theexplain button. When the explain button is clicked, the explaininformation window 1002 may be generated. The explain information window1002 may identify the data associated with the selected object(s) aswell as information (e.g., statistical information) associated with thedata.

In some embodiments, the explain button allows the user to get a sensefor which fields within the selected data fields are responsible for“similarity” of data in the selected nodes and the differentiatingcharacteristics. There can be many ways of scoring the data fields. Theexplain information window 1002 (i.e., the scoring window in FIG. 10) isshown along with the selected nodes. The highest scoring fields maydistinguish variables with respect to the rest of the data.

In one example, the explain information window 1002 indicates that datafrom fields day0-day6 has been selected. The minimum value of the datain all of the fields is 0. The explain information window 1002 alsoindicates the maximum values. For example, the maximum value of all ofthe data associated with the day0 field across all of the points of theselected nodes is 0.353. The average (i.e., mean) of all of the dataassociated with the day0 field across all of the points of the selectednodes is 0.031. The score may be a relative (e.g., normalized) valueindicating the relative function of the filter; here, the score mayindicate the relative density of the data associated with the day0 fieldacross all of the points of the selected nodes. Those skilled in the artwill appreciate that any information regarding the data and/or selectednodes may appear in the explain information window 1002.

It will be appreciated that the data and the interactive visualization1000 may be interacted with in any number of ways. The user may interactwith the data directly to see where the graph corresponds to the data,make changes to the analysis and view the changes in the graph, modifythe graph and view changes to the data, or perform any kind ofinteraction.

FIG. 11 is a flowchart 1100 of functionality of the interactivevisualization in some embodiments. In step 1102, the visualizationengine 322 receives the analysis from the analysis module 320 and graphsnodes as balls and edges as connectors between balls 1202 to createinteractive visualization 900 (see FIG. 9).

In step 1104, the visualization engine 322 determines if the user ishovering a mouse cursor over (or has selected) a ball (i.e., a node). Ifthe user is hovering a mouse cursor over a ball or is selecting a ball,then information may be displayed regarding the data associated with theball. In one example, the visualization engine 322 displays a nodeinformation window 908.

If the visualization engine 322 does not determine that the user ishovering a mouse cursor over (or has selected) a ball, then thevisualization engine 322 determines if the user has selected balls onthe graph (e.g., by clicking on a plurality of balls or drawing a boxaround a plurality of balls). If the user has selected a plurality ofballs on the graph, the visualization engine 322 may highlight theselected balls on the graph in step 1110. The visualization engine 322may also display information regarding the selection (e.g., bydisplaying a selection information window 912). The user may also clickon the explain button 922 to receive more information associated withthe selection (e.g., the visualization engine 322 may display theexplain information window 1002).

In step 1112, the user may save the selection. For example, thevisualization engine 322 may save the underlying data, selected metric,filters, and/or resolution. The user may then access the savedinformation and create a new structure in another interactivevisualization 900 thereby allowing the user to focus attention on asubset of the data.

If the visualization engine 322 does not determine that the user hasselected balls on the graph, the visualization engine 322 may determineif the user selects and drags a ball on the graph in step 1114. If theuser selects and drags a ball on the graph, the visualization engine 322may reorient the selected balls and any connected edges and balls basedon the user's action in step 1116. The user may reorient all or part ofthe structure at any level of granularity.

It will be appreciated that although FIG. 11 discussed the user hoveringover, selecting, and/or dragging a ball, the user may interact with anyobject in the interactive visualization 900 (e.g., the user may hoverover, select, and/or drag an edge). The user may also zoom in or zoomout using the interactive visualization 900 to focus on all or a part ofthe structure (e.g., one or more balls and/or edges). Any number ofactions and operations may be performed using the interactivevisualization 900.

Further, although balls are discussed and depicted in FIGS. 9-11, itwill be appreciated that the nodes may be any shape and appear as anykind of object. Further, although some embodiments described hereindiscuss an interactive visualization being generated based on the outputof algebraic topology, the interactive visualization may be generatedbased on any kind of analysis and is not limited.

For years, researchers have been collecting huge amounts of data onbreast cancer, yet we are still battling the disease. Complexity, ratherthan quantity, is one of the fundamental issues in extracting knowledgefrom data. A topological data exploration and visualization platform mayassist the analysis and assessment of complex data. In variousembodiments, a predictive and visual cancer map generated by thetopological data exploration and visualization platform may assistphysicians to determine treatment options.

In one example, a breast cancer map visualization may be generated basedon the large amount of available information already generated by manyresearchers. Physicians may send biopsy data directly to a cloud-basedserver which may localize a new patient's data within the breast cancermap visualization. The breast cancer map visualization may be annotated(e.g., labeled) such that the physician may view outcomes of patientswith similar profiles as well as different kinds of statisticalinformation such as survival probabilities. Each new data point from apatient may be incorporated into the breast cancer map visualization toimprove accuracy of the breast cancer map visualization over time.

Although the following examples are largely focused on cancer mapvisualizations, it will be appreciated that at least some of theembodiments described herein may apply to any biological condition andnot be limited to cancer and/or disease. For example, some embodiments,may apply to different industries.

FIG. 12 is a flowchart for generating a cancer map visualizationutilizing biological data of a plurality of patients in someembodiments. In various embodiments, the processing of data anduser-specified options is motivated by techniques from topology and, insome embodiments, algebraic topology. As discussed herein, thesetechniques may be robust and general. In one example, these techniquesapply to almost any kind of data for which some qualitative idea of“closeness” or “similarity” exists. It will be appreciated that theimplementation of techniques described herein may apply to any level ofgenerality.

In various embodiments, a cancer map visualization is generated usinggenomic data linked to clinical outcomes (i.e., medical characteristics)which may be used by physicians during diagnosis and/or treatment.Initially, publicly available data sets may be integrated to constructthe topological map visualizations of patients (e.g., breast cancerpatients). It will be appreciated that any private, public, orcombination of private and public data sets may be integrated toconstruct the topological map visualizations. A map visualization may bebased on biological data such as, but not limited to, gene expression,sequencing, and copy number variation. As such, the map visualizationmay comprise many patients with many different types of collected data.Unlike traditional methods of analysis where distinct studies of breastcancer appear as separate entities, the map visualization may fusedisparate data sets while utilizing many datasets and data types.

In various embodiments, a new patient may be localized on the mapvisualization. With the map visualization for subtypes of a particulardisease and a new patient diagnosed with the disease, point(s) may belocated among the data points used in computing the map visualization(e.g., nearest neighbor) which is closest to the new patient point. Thenew patient may be labeled with nodes in the map visualizationcontaining the closest neighbor. These nodes may be highlighted to givea physician the location of the new patient among the patients in thereference data set. The highlighted nodes may also give the physicianthe location of the new patient relative to annotated disease subtypes.

The visualization map may be interactive and/or searchable in real-timethereby potentially enabling extended analysis and providing speedyinsight into treatment.

In step 1202, biological data and clinical outcomes of previous patientsmay be received. The clinical outcomes may be medical characteristics.Biological data is any data that may represent a condition (e.g., amedical condition) of a person. Biological data may include any healthrelated, medical, physical, physiological, pharmaceutical dataassociated with one or more patients. In one example, biological datamay include measurements of gene expressions for any number of genes. Inanother example, biological data may include sequencing information(e.g., RNA sequencing).

In various embodiments, biological data for a plurality of patients maybe publicly available. For example, various medical health facilitiesand/or public entities may provide gene expression data for a variety ofpatients. In addition to the biological data, information regarding anynumber of clinical outcomes, treatments, therapies, diagnoses and/orprognoses may also be provided. Those skilled in the art will appreciatethat any kind of information may be provided in addition to thebiological data.

The biological data, in one example, may be similar to data S asdiscussed with regard to step 802 of FIG. 8. The biological data mayinclude ID fields that identify patients and data fields that arerelated to the biological information (e.g., gene expressionmeasurements).

FIG. 13 is an example data structure 1300 including biological data 1304a-1304 y for a number of patients 1308 a-1308 n that may be used togenerate the cancer map visualization in some embodiments. Column 1302represents different patient identifiers for different patients. Thepatient identifiers may be any identifier.

At least some biological data may be contained within gene expressionmeasurements 1304 a-1304 y. In FIG. 13, “y” represents any number. Forexample, there may be 50,000 or more separate columns for different geneexpressions related to a single patient or related to one or moresamples from a patient. It will be appreciated that column 1304 a mayrepresent a gene expression measurement for each patient (if any forsome patients) associated with the patient identifiers in column 1302.The column 1304 b may represent a gene expression measurement of one ormore genes that are different than that of column 1304 a. As discussed,there may be any number of columns representing different geneexpression measurements.

Column 1306 may include any number of clinical outcomes, prognoses,diagnoses, reactions, treatments, and/or any other informationassociated with each patient. All or some of the information containedin column 1306 may be displayed (e.g., by a label or an annotation thatis displayed on the visualization or available to the user of thevisualization via clicking) on or for the visualization.

Rows 1308 a-1308 n each contains biological data associated with thepatient identifier of the row. For example, gene expressions in row 1308a are associated with patient identifier P1. As similarly discussed withregard to “y” herein, “n” represents any number. For example, there maybe 100,000 or more separate rows for different patients.

It will be appreciated that there may be any number of data structuresthat contain any amount of biological data for any number of patients.The data structure(s) may be utilized to generate any number of mapvisualizations.

In step 1204, the analysis server may receive a filter selection. Insome embodiments, the filter selection is a density estimation function.It will be appreciated that the filter selection may include a selectionof one or more functions to generate a reference space.

In step 1206, the analysis server performs the selected filter(s) on thebiological data of the previous patients to map the biological data intoa reference space. In one example, a density estimation function, whichis well known in the art, may be performed on the biological data (e.g.,data associated with gene expression measurement data 1304 a-1304 y) torelate each patient identifier to one or more locations in the referencespace (e.g., on a real line).

In step 1208, the analysis server may receive a resolution selection.The resolution may be utilized to identify overlapping portions of thereference space (e.g., a cover of the reference space R) in step 1210.

As discussed herein, the cover of R may be a finite collection of opensets (in the metric of R) such that every point in R lies in at leastone of these sets. In various examples, R is k-dimensional Euclideanspace, where k is the number of filter functions. Those skilled in theart will appreciate that the cover of the reference space R may becontrolled by the number of intervals and the overlap identified in theresolution (e.g., see FIG. 7). For example, the more intervals, thefiner the resolution in S (e.g., the similarity space of the receivedbiological data)—that is, the fewer points in each S(d), but the moresimilar (with respect to the filters) these points may be. The greaterthe overlap, the more times that clusters in S(d) may intersect clustersin S(e)—this means that more “relationships” between points may appear,but, in some embodiments, the greater the overlap, the more likely thataccidental relationships may appear.

In step 1212, the analysis server receives a metric to cluster theinformation of the cover in the reference space to partition S(d). Inone example, the metric may be a Pearson Correlation. The clusters mayform the groupings (e.g., nodes or balls). Various cluster means may beused including, but not limited to, a single linkage, average linkage,complete linkage, or k-means method.

As discussed herein, in some embodiments, the analysis module 320 maynot cluster two points unless filter values are sufficiently “related”(recall that while normally related may mean “close,” the cover mayimpose a much more general relationship on the filter values, such asrelating two points s and t if ref(s) and ref(t) are sufficiently closeto the same circle in the plane where ref( ) represents one or morefilter functions). The output may be a simplicial complex, from whichone can extract its 1-skeleton. The nodes of the complex may be partialclusters, (i.e., clusters constructed from subsets of S specified as thepreimages of sets in the given covering of the reference space R).

In step 1214, the analysis server may generate the visualization mapwith nodes representing clusters of patient members and edges betweennodes representing common patient members. In one example, the analysisserver identifies nodes which are associated with a subset of thepartition elements of all of the S(d) for generating an interactivevisualization.

As discussed herein, for example, suppose that S={1, 2, 3, 4}, and thecover is C₁, C₂, C₃. Suppose cover C₁ contains {1, 4}, C₂ contains{1,2}, and C₃ contains {1,2,3,4}. If 1 and 2 are close enough to beclustered, and 3 and 4 are, but nothing else, then the clustering forS(1) may be {1}, {4}, and for S(2) it may be {1,2}, and for S(3) it maybe {1,2}, {3,4}. So the generated graph has, in this example, at mostfour nodes, given by the sets {1}, {4}, {1, 2}, and {3, 4} (note that{1, 2} appears in two different clusterings). Of the sets of points thatare used, two nodes intersect provided that the associated node setshave a non-empty intersection (although this could easily be modified toallow users to require that the intersection is “large enough” either inabsolute or relative terms).

As a result of clustering, member patients of a grouping may sharebiological similarities (e.g., similarities based on the biologicaldata).

The analysis server may join clusters to identify edges (e.g.,connecting lines between nodes). Clusters joined by edges (i.e.,interconnections) share one or more member patients. In step 1216, adisplay may display a visualization map with attributes based on theclinical outcomes contained in the data structures (e.g., see FIG. 13regarding clinical outcomes). Any labels or annotations may be utilizedbased on information contained in the data structures. For example,treatments, prognoses, therapies, diagnoses, and the like may be used tolabel the visualization. In some embodiments, the physician or otheruser of the map visualization accesses the annotations or labels byinteracting with the map visualization.

The resulting cancer map visualization may reveal interactions andrelationships that were obscured, untested, and/or previously notrecognized.

FIG. 14 is an example visualization displaying the cancer mapvisualization 1400 in some embodiments. The cancer map visualization1400 represents a topological network of cancer patients. The cancer mapvisualization 1400 may be based on publicly and/or privately availabledata.

In various embodiments, the cancer map visualization 1400 is createdusing gene expression profiles of excised tumors. Each node (i.e., ballor grouping displayed in the map visualization 1400) contains a subsetof patients with similar genetic profiles.

As discussed herein, one or more patients (i.e., patient members of eachnode or grouping) may occur in multiple nodes. A patient may share asimilar genetic profile with multiple nodes or multiple groupings. Inone example, of 50,000 different gene expressions of the biologicaldata, multiple patients may share a different genetic profiles (e.g.,based on different gene expression combinations) with differentgroupings. When a patient shares a similar genetic profile withdifferent groupings or nodes, the patient may be included within thegroupings or nodes.

The cancer map visualization 1400 comprises groupings andinterconnections that are associated with different clinical outcomes.All or some of the clinical outcomes may be associated with thebiological data that generated the cancer map visualization 1400. Thecancer map visualization 1400 includes groupings associated withsurvivors 1402 and groupings associated with non-survivors 1404. Thecancer map visualization 1400 also includes different groupingsassociated with estrogen receptor positive non-survivors 1406, estrogenreceptor negative non-survivors 1408, estrogen receptor positivesurvivors 1410, and estrogen receptor negative survivors 1412.

In various embodiments, when one or more patients are members of two ormore different nodes, the nodes are interconnected by an edge (e.g., aline or interconnection). If there is not an edge between the two nodes,then there are no common member patients between the two nodes. Forexample, grouping 1414 shares at least one common member patient withgrouping 1418. The intersection of the two groupings is represented byedge 1416. As discussed herein, the number of shared member patients ofthe two groupings may be represented in any number of ways includingcolor of the interconnection, color of the groupings, size of theinterconnection, size of the groupings, animations of theinterconnection, animations of the groupings, brightness, or the like.In some embodiments, the number and/or identifiers of shared memberpatients of the two groupings may be available if the user interactswith the groupings 1414 and/or 1418 (e.g., draws a box around the twogroupings and the interconnection utilizing an input device such as amouse).

In various embodiments, a physician, on obtaining some data on a breasttumor, direct the data to an analysis server (e.g., analysis server 208over a network such as the Internet) which may localize the patientrelative to one or more groupings on the cancer map visualization 1400.The context of the cancer map visualization 1400 may enable thephysician to assess various possible outcomes (e.g., proximity ofrepresentation of new patient to the different associations of clinicaloutcomes).

FIG. 15 is a flowchart of for positioning new patient data relative to acancer map visualization in some embodiments. In step 1502, newbiological data of a new patient is received. In various embodiments, aninput module 314 of an analysis server (e.g., analysis server 208 ofFIGS. 1 and 2) may receive biological data of a new patient from aphysician or medical facility that performed analysis of one or moresamples to generate the biological data. The biological data may be anydata that represents a biological data of the new patient including, forexample, gene expressions, sequencing information, or the like.

In some embodiments, the analysis server 208 may comprise a new patientdistance module and a location engine. In step 1504, the new patientdistance module determines distances between the biological data of eachpatient of the cancer map visualization 1600 and the new biological datafrom the new patient. For example, the previous biological data that wasutilized in the generation of the cancer map visualization 1600 may bestored in mapped data structures. Distances may be determined betweenthe new biological data of the new patient and each of the previouspatient's biological data in the mapped data structure.

It will be appreciated that distances may be determined in any number ofways using any number of different metrics or functions. Distances maybe determined between the biological data of the previous patients andthe new patients. For example, a distance may be determined between afirst gene expression measurement of the new patient and each (or asubset) of the first gene expression measurements of the previouspatients (e.g., the distance between G1 of the new patient and G1 ofeach previous patient may be calculated). Distances may be determinedbetween all (or a subset of) other gene expression measurements of thenew patient to the gene expression measurements of the previouspatients.

In various embodiments, a location of the new patient on the cancer mapvisualization 1600 may be determined relative to the other memberpatients utilizing the determined distances.

In step 1506, the new patient distance module may compare distancesbetween the patient members of each grouping to the distances determinedfor the new patient. The new patient may be located in the grouping ofpatient members that are closest in distance to the new patient. In someembodiments, the new patient location may be determined to be within agrouping that contains the one or more patient members that are closestto the new patient (even if other members of the grouping have longerdistances with the new patient). In some embodiments, this step isoptional.

In various embodiments, a representative patient member may bedetermined for each grouping. For example, some or all of the patientmembers of a grouping may be averaged or otherwise combined to generatea representative patient member of the grouping (e.g., the distancesand/or biological data of the patient members may be averaged oraggregated). Distances may be determined between the new patientbiological data and the averaged or combined biological data of one ormore representative patient members of one or more groupings. Thelocation engine may determine the location of the new patient based onthe distances. In some embodiments, once the closest distance betweenthe new patient and the representative patient member is found,distances may be determined between the new patient and the individualpatient members of the grouping associated with the closestrepresentative patient member.

In optional step 1508, a diameter of the grouping with the one or moreof the patient members that are closest to the new patient (based on thedetermined distances) may be determined. In one example, the diametersof the groupings of patient members closest to the new patient arecalculated. The diameter of the grouping may be a distance between twopatient members who are the farthest from each other when compared tothe distances between all patient members of the grouping. If thedistance between the new patient and the closest patient member of thegrouping is less than the diameter of the grouping, the new patient maybe located within the grouping. If the distance between the new patientand the closest patient member of the grouping is greater than thediameter of the grouping, the new patient may be outside the grouping(e.g., a new grouping may be displayed on the cancer map visualizationwith the new patient as the single patient member of the grouping). Ifthe distance between the new patient and the closest patient member ofthe grouping is equal to the diameter of the grouping, the new patientmay be placed within or outside the grouping.

It will be appreciated that the determination of the diameter of thegrouping is not required in determining whether the new patient locationis within or outside of a grouping. In various embodiments, adistribution of distances between member patients and between memberpatients and the new patient is determined. The decision to locate thenew patient within or outside of the grouping may be based on thedistribution. For example, if there is a gap in the distribution ofdistances, the new patient may be separated from the grouping (e.g., asa new grouping). In some embodiments, if the gap is greater than apreexisting threshold (e.g., established by the physician, other user,or previously programmed), the new patient may be placed in a newgrouping that is placed relative to the grouping of the closest memberpatients. The process of calculating the distribution of distances ofcandidate member patients to determine whether there may be two or moregroupings may be utilized in generation of the cancer map visualizationfurther described herein (e.g., in the process as described with regardto FIG. 12). It will be appreciated that there may be any number of waysto determine whether a new patient should be included within a groupingof other patient members.

In step 1510, the location engine determines the location of the newpatient relative to the member patients and/or groupings of the cancermap visualization. The new location may be relative to the determineddistances between the new patient and the previous patients. Thelocation of the new patient may be part of a previously existinggrouping or may form a new grouping.

In some embodiments, the location of the new patient with regard to thecancer map visualization may be performed locally to the physician. Forexample, the cancer map visualization 1400 may be provided to thephysician (e.g., via a digital device). The physician may load the newpatient's biological data locally and the distances may be determinedlocally or via a cloud-based server. The location(s) associated with thenew patient may be overlaid on the previously existing cancer mapvisualization either locally or remotely.

It will be appreciated that, in some embodiments, the previous state ofthe cancer map visualization (e.g., cancer map visualization 1400) maybe retained or otherwise stored and a new cancer map visualizationgenerated utilizing the new patient biological data (e.g., in a methodsimilar to that discussed with regard to FIG. 12). The newly generatedmap may be compared to the previous state and the differences may behighlighted thereby, in some embodiments, highlighting the location(s)associated with the new patient. In this way, distances may be not becalculated as described with regard to FIG. 15, but rather, the processmay be similar to that as previously discussed.

FIG. 16 is an example visualization displaying the cancer map includingpositions for three new cancer patients in some embodiments. The cancermap visualization 1400 comprises groupings and interconnections that areassociated with different clinical outcomes as discussed with regard toFIG. 14. All or some of the clinical outcomes may be associated with thebiological data that generated the cancer map visualization 1400. Thecancer map visualization 1400 includes different groupings associatedwith survivors 1402, groupings associated with non-survivors 1404,estrogen receptor positive non-survivors 1406, estrogen receptornegative non-survivors 1408, estrogen receptor positive survivors 1410,and estrogen receptor negative survivors 1412.

The cancer map visualization 1400 includes three locations for three newbreast cancer patients. The breast cancer patient location 1602 isassociated with the clinical outcome of estrogen receptor positivesurvivors. The breast cancer patient location 1604 is associated withthe clinical outcome of estrogen receptor negative survivors.Unfortunately, breast cancer patient location 1606 is associated withestrogen receptor negative non-survivors. Based on the locations, aphysician may consider different diagnoses, prognoses, treatments, andtherapies to maintain or attempt to move the breast cancer patient to adifferent location utilizing the cancer map visualization 1400.

In some embodiments, the physician may assess the underlying biologicaldata associated with any number of member patients of any number ofgroupings to better understand the genetic similarities and/ordissimilarities. The physician may utilize the information to makebetter informed decisions.

The patient location 1604 is highlighted on the cancer map visualization1400 as active (e.g., selected by the physician). It will be appreciatedthat the different locations may be of any color, size, brightness,and/or animated to highlight the desired location(s) for the physician.Further, although only one location is identified for three differentbreast cancer patients, any of the breast cancer patients may havemultiple locations indicating different genetic similarities.

It will be appreciated that the cancer map visualization 1400 may beupdated with new information at any time. As such, as new patients areadded to the cancer map visualization 1400, the new data updates thevisualization such that as future patients are placed in the map, themap may already include the updated information. As new informationand/or new patient data is added to the cancer map visualization 1400,the cancer map visualization 1400 may improve as a tool to better informphysicians or other medical professionals.

In various embodiments, the cancer map visualization 1400 may trackchanges in patients over time. For example, updates to a new patient maybe visually tracked as changes in are measured in the new patient'sbiological data. In some embodiments, previous patient data is similarlytracked which may be used to determine similarities of changes based oncondition, treatment, and/or therapies, for example. In variousembodiments, velocity of change and/or acceleration of change of anynumber of patients may be tracked over time using or as depicted on thecancer map visualization 1400. Such depictions may assist the treatingphysician or other personnel related to the treating physician to betterunderstand changes in the patient and provide improved, current, and/orupdated diagnoses, prognoses, treatments, and/or therapies.

FIG. 17 is a flowchart of utilization the visualization and positioningof new patient data in some embodiments. In various embodiments, aphysician may collect amounts of genomic information from tumors removedfrom a new patient, input the data (e.g., upload the data to an analysisserver), and receive a map visualization with a location of the newpatient. The new patient's location within the map may offer thephysician new information about the similarities to other patients. Insome embodiments, the map visualization may be annotated so that thephysician may check the outcomes of previous patients in a given regionof the map visualization are distributed and then use the information toassist in decision-making for diagnosis, treatment, prognosis, and/ortherapy.

In step 1702, a medical professional or other personnel may remove asample from a patient. The sample may be of a tumor, blood, or any otherbiological material. In one example, a medical professional performs atumor excision. Any number of samples may be taken from a patient.

In step 1704, the sample(s) may be provided to a medical facility todetermine new patient biological data. In one example, the medicalfacility measures genomic data such as gene expression of a number ofgenes or protein levels.

In step 1706, the medical professional or other entity associated withthe medical professional may receive the new patient biological databased on the sample(s) from the new patient. In one example, a physicianmay receive the new patient biological data. The physician may provideall or some of the new patient biological data to an analysis serverover the Internet (e.g., the analysis server may be a cloud-basedserver). In some embodiments, the analysis server is the analysis server208 of FIG. 2. In some embodiments, the medical facility that determinesthe new patient biological data provides the biological data in anelectronic format which may be uploaded to the analysis server. In someembodiments, the medical facility that determines the new patientbiological data (e.g., the medical facility that measures the genomicdata) provide the biological data to the analysis server at the requestof the physician or others associated with the physician. It will beappreciated that the biological data may be provided to the analysisserver in any number of ways.

The analysis server may be any digital device and may not be limited toa digital device on a network. In some embodiments, the physician mayhave access to the digital device. For example, the analysis server maybe a table, personal computer, local server, or any other digitaldevice.

Once the analysis server receives the biological data of the new patient(e.g., the new patient biological data may be uploaded to the analysisserer in step 1708), the new patient may be localized in the mapvisualization and the information may be sent back to the physician instep 1710. The visualization may be a map with nodes representingclusters of previous patient members and edges between nodesrepresenting common patient members. The visualization may furtherdepict one or more locations related to the biological data of the newpatient.

The map visualization may be provided to the physician or otherassociated with the physician in real-time. For example, once thebiological data associated with the new patient is provided to theanalysis server, the analysis server may provide the map visualizationback to the physician or other associated with the physician within areasonably short time (e.g., within seconds or minutes). In someembodiments, the physician may receive the map visualization over anytime.

The map visualization may be provided to the physician in any number ofways. For example, the physician may receive the map visualization overany digital device such as, but not limited to, an office computer,IPad, tablet device, media device, smartphone, e-reader, or laptop.

In step 1712, the physician may assess possible different clinicaloutcomes based on the map visualization. In one example, the map-aidedphysician may make decisions on therapy and treatments depending onwhere the patient lands on the visualization (e.g., survivor ornon-survivor). The map visualization may include annotations or labelsthat identify one or more sets of groupings and interconnections asbeing associated with one or more clinical outcomes. The physician mayassess possible clinical outcomes based on the position(s) on the mapassociated with the new patient.

FIG. 18 is a block diagram of an exemplary digital device 1800. Thedigital device 1800 comprises a processor 1802, a memory system 1804, astorage system 1806, a communication network interface 1808, an I/Ointerface 1810, and a display interface 1812 communicatively coupled toa bus 1814. The processor 1802 may be configured to execute executableinstructions (e.g., programs). In some embodiments, the processor 1802comprises circuitry or any processor capable of processing theexecutable instructions.

The memory system 1804 is any memory configured to store data. Someexamples of the memory system 1804 are storage devices, such as RAM orROM. The memory system 1804 can comprise the ram cache. In variousembodiments, data is stored within the memory system 1804. The datawithin the memory system 1804 may be cleared or ultimately transferredto the storage system 1806.

The storage system 1806 is any storage configured to retrieve and storedata. Some examples of the storage system 1806 are flash drives, harddrives, optical drives, and/or magnetic tape. In some embodiments, thedigital device 1800 includes a memory system 1804 in the form of RAM anda storage system 1806 in the form of flash data. Both the memory system1804 and the storage system 1806 comprise computer readable media whichmay store instructions or programs that are executable by a computerprocessor including the processor 1802.

The communication network interface (com. network interface) 1808 can becoupled to a data network (e.g., communication network 204) via the link1816. The communication network interface 1808 may support communicationover an Ethernet connection, a serial connection, a parallel connection,or an ATA connection, for example. The communication network interface1808 may also support wireless communication (e.g., 1802.11 a/b/g/n,WiMAX). It will be apparent to those skilled in the art that thecommunication network interface 1808 can support many wired and wirelessstandards.

The optional input/output (I/O) interface 1810 is any device thatreceives input from the user and output data. The optional displayinterface 1812 is any device that may be configured to output graphicsand data to a display. In one example, the display interface 1812 is agraphics adapter.

It will be appreciated that the hardware elements of the digital device1800 are not limited to those depicted in FIG. 18. A digital device 1800may comprise more or less hardware elements than those depicted.Further, hardware elements may share functionality and still be withinvarious embodiments described herein. In one example, encoding and/ordecoding may be performed by the processor 1802 and/or a co-processorlocated on a GPU.

FIG. 19 shows example a landmark module 1900 configured to identifylandmark points that approximate or represent a larger collection ofdata points in accordance with various embodiments. In this example,landmark module 1900 comprises landmark selection module 1902, adistance calculation module 1904, a landmark distance identificationmodule 1906, a landmark distance storage module 1908, a landmarkdistance comparison module 1910, and a landmark assignment module 1912.

The landmark selection module 1902 may be configured to randomly selecta first subset of the data points to assign as an initial set oflandmark points. For example, the landmark selection module 1902 mayselect an initial set of points from the finite metric space as alandmark set L. It will be appreciated that the landmark selectionmodule 1902 may select points pseudo-randomly (e.g., randomly within thebounds of software or computer implementation) and/or in combinationwith other methods (e.g., randomly within portions of the finite metricspace or based, in part, on density of information). Landmark selectionmodule 1902 may select points in any number of ways (e.g., the landmarkselection module 1902 may select points based on any methodology and/ormay not select points randomly).

The distance calculation module 1904 may be configured to calculate thedistances between a respective non-landmark data point and each landmarkpoint in the finite reference space. In some embodiments, the distancecalculation module 1904 stores some or all of the information for lateruse.

The landmark distance identification module 1906 may be configured toidentify the shortest distance from among the distances between therespective non-landmark data point and each landmark. The shortestdistance between a non-landmark data point and a landmark data point mayindicate the closest landmark to that particular non-landmark datapoint.

The landmark distance storage module 1908 may be configured to store theshortest data point distance for the respective non-landmark data pointas a landmark distance for that data point. The landmark distancecomparison module 1910 may be configured to determine a longest landmarkdistance from among the shortest distances (e.g., stored by the landmarkdistance storage module 1908) to a nearest landmark for each data point.

The landmark assignment module 1912 may be configured to add a datapoint associated with the longest landmark distance to the initial setof landmark points thereby adding a new landmark and creating a new setof landmark points.

As described herein, the landmarks (L) are a subset of the collectiondata points in the finite metric space. The landmarks may be chosen suchthat the subset is representative of or to approximate the receiveddata. In some embodiments, the landmarks are chosen to reflect both the“average” and “extreme” behavior of the data points in the space and,thus, analytics and other operations performed on the landmark set as anapproximation of the behavior of the whole metric space (X). In someembodiments, the landmark points may be used as a means of increasingscale and performance when working with a large collection of data byonly operating on a subset of a space.

FIG. 20 is a flow chart 2000 depicting an example method for generatinga set of landmark points from a data set in some embodiments. Thefollowing discussion regarding the steps in FIG. 20 will be describedwith references to FIGS. 21A-D and FIG. 22A-C. In step 2002, thelandmark selection module 1902 receives a set of data points defining afinite metric space. For example, receiving data may include landmarkselection module 1902 accessing a data structure containing a very largevolume of multidimensional data, as shown in FIG. 21A.

FIG. 21A shows example metric space 2100 containing data in accordancewith various embodiments. Since the amount of data shown in metric space2100 handled by the methods and algorithms discussed herein may be large(e.g., on the order of 200 million+ data points), subset 2102 of metricspace 2100 will be used for discussion purposes. Accordingly, FIG. 21Bshows subset 2102 composed of individual data points 2104 in accordancewith some embodiments.

At step 2004, landmark selection module 2902 selects a random subset ofindividual data points 2104 as a first set (e.g., an initial set) oflandmark points. To illustrate this step, FIG. 21C shows example randomlandmarks R₁, R₂, R₃, and R₄ that have been randomly selected as initiallandmarks. Since metric space 2100 is large (e.g., 200 million+ datapoints), points selected at random tend to be located in high densityareas, which is a benefit when attempting to choose a subset of pointsthat represent the characteristics of the larger space. For example, fora metric space of approximately 200 million data points, the number ofrandomly selected landmark points could be approximately 5,000 points.Thus, the probability that a significant portion of the randomlyselected landmarks may end up being outliers, for example, may be quitelow and the randomly selected landmarks end up being located in higherdensity data point regions.

At step 2006, for each non-landmark point, the distance calculationmodule 1904 calculates distances between that particular non-landmarkpoint and each landmark point. As used herein, the distances betweenlandmark points and individual data points 2104 are referred to as datapoint distances. Accordingly, FIG. 21D shows lines corresponding to datapoint distances to each landmark for three points (P₁, P₂, and P₃). Itshould be appreciated that, in various embodiments, the data pointdistances for all other points other than P₁, P₂, and P₃ and thelandmarks are also calculated, but of clarity and illustrative purposes,the lines shown in FIG. 21D have only been drawn for P₁, P₂, and P₃.Accordingly, in this example, each distance between P₁ and R₁, R₂, R₃,and R₄ is calculated, each distance between P₂ and R₁, R₂, R₃, and R₄ iscalculated, etc. until the distances between each non-landmark point andall the landmarks are calculated. FIGS. 22A and 22B show this process inmore detail.

FIG. 22A shows example data point distances between point P₁ and randomlandmarks R₁, R₂, R₃, and R₄. In this example, distance d₁ between P₁and R₁ is 3, distance d₂ between P₁ and R₂ is 5, distance d₃ between P₁and R₃ is 7, and distance d₄ between P₁ and R₄ is 6. In variousembodiments, the landmark distance for a respective non-landmark pointis defined as the shortest distance to its nearest landmark or theshortest data point distance. In this example, distances d₁, d₂, d₃, andd₄ are compared to each other to determine which is the shortestdistance to a landmark from P₁. In this example, distance d₁, between P₁and R₁, is the shortest distance and, thus, defined as landmark distance2202 for P₁. Accordingly, R₁ is the closest landmark to P₁ withcorresponding landmark distance 2202 (i.e., d₁=3).

Similarly, FIG. 22B shows example distances between point P₂ and randomlandmarks R₁, R₂, R₃, and R₄. In this example, distance d₅ between P₂and R₁ is 5, distance d₆ between P₂ and R₂ is 5, distance d₇ between P₂and R₃ is 9, and distance d₈ between P₂ and R₄ is 8. As above, distancesd₅, d₆, d₇, and d₈ are compared to each other to determine which is theshortest distance to P₂'s nearest landmark, which is distance d₅.Accordingly, distance d₅ between P₂ and R₁ is landmark distance 2204.Thus, R₁ is also the closest landmark to P₂ at landmark distance 2204(i.e., d₅=5), in this example.

Accordingly, the distance calculations described in FIGS. 22A and 22Bare, thus, calculated for P₃ and every other non-landmark point inmetric space 2100 and the distance calculations may be stored. Forexample, FIG. 22C shows an example table 2250 wherein distances for eachpoint are stored. Although FIG. 22C depicts a table, it will beappreciated that any data structure(s) or combination of datastructure(s) may be utilized. Further, although table 2250 includes alldistances from P1 to each landmark, it will be appreciated that, in someembodiments, a subset of the distances may be stored. In one example,only the shortest distance between P1 and the closest landmark may bestored.

Further, in this example, only the distances for points P₁ and P₂ areshown, but it should be appreciated that such a table or array wouldinclude distances for each non-landmark point. Thus, in one embodiment,table 2250 stores the distances for each point to each landmark inmetric space 2100. From these distances, a landmark distance (e.g.,shortest distance to a nearest landmark) for each point may beidentified and compared to generate a second set of landmark points.This process is discussed further with respect to FIGS. 23A-23D.

At step 2008, landmark distance identification module 1906 identifiesthe shortest data point distance from among the data point distances.FIG. 23A shows example landmark distances for points P₁, P₂, and P₃ tolandmark R₁ which can be used to demonstrate the selection of additionallandmark points. For example, landmark distance identification module1906 determines for each point which landmark point is the closestlandmark point for that respective point. This may include, for example,comparing the distance values d_(n) from table 2250 for each point todetermine which distance d_(n) is the shortest. Accordingly, in thisexample, the shortest between a landmark and P₁ is 3 (i.e., between P₁and landmark point R₁) and the shortest distance to a landmark pointfrom P₂ is 5 which is also to landmark point R₁.

Such an operation may use an indexable state for X (i.e., points such asP₁, P₂, and P₃ in metric space 2100), an indexable array for L (e.g.,L[1] is the index in X of the l′th landmark) where each random landmarkpoint R_(n) and subsequently determined landmark point is in L, anddClosest[x] which records the shortest distance between X[x] (i.e., P₁,P₂, P₃, etc.) and a respective closest landmark point, and inL[ ] withis true if x is in L.

At step 2010, landmark distance storage module 1908 stores the shortestdistance from each non-landmark point to a landmark point (or thedistance to the nearest landmark) in an array. FIG. 23 B shows exampleshortest distances from each non-landmark point to each landmark point.In FIG. 23B, table 2350 contains the shortest distances between eachdata point P₁, P₂, and P₃ and its closest landmark, respectively.

In various embodiments, for each non-landmark point, the closestlandmark point is identified. As a result, a list of non-landmark pointsthat identify the same landmark point as the closest landmark point maybe identified. For example, for each such landmark point, a table suchas table 2350 may be generated that identifies the non-landmark pointsthat identify the same particular landmark point as being closest. Thetable 2350 may further identify distances between those non-landmarkpoints and the same particular landmark point. In this example, table2350 may contain the shortest distances between data points P₁, P₂, andP₃ and landmark point R₁. data point and only one landmark R₁

At step 2012, landmark distance comparison module 1910 determines alongest landmark distance from among each of the shortest data pointdistances (or a longest landmark distance) from among each of thelandmark distances. For example, returning to FIG. 23A, random landmarkpoint R₁ is the landmark nearest to points P₁, P₂, and P₃ and, thus, thelandmark distance l_(n) (i.e., the distance to a nearest landmark) foreach of these points is its respective distance to R₁, which may bestored in table 2350. Thus, in this example, the landmark distance forP₁ is l₁=3, the landmark distance for P₂ is l₂=5, and the landmarkdistance for P₃ is l₃=4. Accordingly, landmark distance comparisonmodule 1910 compares these distances to identify the longest distancewhich, in this example, is l₂=5 shown circled in FIG. 23B, belonging topoint P₂.

Thus, with the longest landmark distance, P₂ is maximally far away fromthe random landmarks relative to the other non-landmark points and, atstep 2014, landmark assignment module 1912 adds P₂ to the set of randomlandmark points (or seed landmarks) to generate a new set of landmarkpoints. Thus, there is an initial set of randomly selected landmarkpoints (R) and max-min landmark points (MM) calculated along the way aresubsequently added to R to generate a set of landmarks (L). Accordingly,FIG. 23C shows point P₂ as new MM landmark point L₁.

In various embodiments, this process may start over to identify and adda second most maximally far away point to the set of landmark pointsafter L₁ has been added to the initial set of randomly selected landmarkpoints (R). Thus, steps 2002 to 2014 can be repeated with L₁ includedinto the set of landmark points (L) when determining the landmarkdistances for each point. Accordingly, FIG. 23D shows subset 2102 withL₁ as a new landmark where the distances between various points havebeen calculated. In this example, R₁ is no longer the closest landmarkto points P₁ and P₃ with the inclusion of L₁ and L₂. For example, P₁ isnow a distance d_(1′)=2 from its nearest landmark L₁ and P₃, whosenearest landmark is also L₁, is now a distance d_(3′)=2 from L₁.Further, as shown in FIG. 23D, the distance d_(4′)=3 between point P₄and R₁ and the distance d_(5′)=4 between point P₄ and newly added MMlandmark point L₂ since d_(5′) is larger than d_(4′), d_(3′), andd_(1′).

In one example, a method for generating a set of landmark points canutilize a process called PROCESS_x_AND_l(X,l), for example, thatdetermines the distances between each point and each landmark point,identifies the closest landmark for each point (dClosest[ ]), andupdates an array of dClosest[ ] for each point. Subsequently, a processcalled FIND_NEXT_L(1) can add a new MM landmark at l to the set oflandmarks (L). For example, PROCESS_x_AND_l(x,l) can be implemented asfollows:

double dist = distance(x, L[l]); if (dist < dClosest[x]) dClosest[x] =dist;FIND_NEXT_L(1) can be implemented as follows:

double closestD = −Double.MAX_VALUE; for (int x = 0; x < |X|; x++) {  if(!inL[x] && (dClosest[x] > closestD)) { closestD = dClosest[x]; L[l] =x;

Thus, referring back to FIG. 23D, the method for generating a set oflandmark points can proceed by first selecting random landmarks R₁, R₂,R₃, and R₄ and, thereafter, successively calling PROCESS_x_AND_l(x,l)for each point in metric space 2100 (e.g., each x in X on every l in L).Accordingly, a first portion of a method for generating a set oflandmark points can be implemented as follows:

for l = 0, l < |R| l++ do for x = 0, x < |X|, x++ doPROCESS_x_AND_l(x,l)

Once the first portion is completed, the remaining landmark points canbe looped over one at a time to find the next MM landmark in a secondportion of the method:

for l = |R|, l < |L|, l++ do FIND_NEXT_L(l) for x = 0, x < |X|, x++ doPROCESS_x_AND_l(x,l) done

If the landmark selection process is improperly implemented, it can beinefficient for large spaces. For example, the |L|×|X| matrix can behuge and, if the distance calculations are not ordered properly, thecomputation can page wildly. For example, as described above, thelandmark selection process iterates |L| (i.e., the number of landmarkpoints) times over the data X (i.e., the number of data points) ofmetric space 2100. If the data space X does not fit into availablememory on a computer system, the data in X gets read repeatedly fromdisc, with slow results.

It will be appreciated that landmarks may be used instead of an entiredata set for analysis. The landmark set may approximate the behavior ofa larger data set thereby allowing analysis of the landmark set forcomputational efficiency and speed.

The landmark process may be used at many different stages in topologicalanalysis (examples of topological analysis are described herein). Forexample, landmarks of data points mapped to a reference space may beidentified. The landmark set may then be utilized to create avisualization as also described herein. In one example, as discussedregarding FIG. 8, the input module 314 may receive data (e.g., data S).In one example, a user identifies a data structure and then identifiesID and data fields. Data S may be based on the information within the IDand data fields. It will be appreciated that data S may be a finitemetric space, or a generalization thereof, such as a graph or weightedgraph.

The input module 314 may generate reference space R. In one example,reference space R may be a well-known metric space (e.g., such as thereal line). The reference space R may be defined by the user. Theanalysis module 320 may generate a map ref( ) from S into R. The mapref( ) from S into R may be called the “reference map.”

A landmark set of data points may be determined using methods describedherein. The landmark set of data points may be a subset of the datapoints mapped into the reference space. For example, a first subset ofthe data points in the map may be selected to generate an initial set oflandmarks. Each data point of the first subset may define a landmarkpoint.

As discussed herein, for each non-landmark data point, first data pointdistances between a respective non-landmark data point and each landmarkpoint of the initial set of landmarks may be calculated, a firstshortest data point distance from among the first data point distancesbetween the respective non-landmark data point and each landmark pointof the initial set of landmarks may be identified, and the firstshortest data point distance as a first landmark distance for therespective non-landmark data point may be stored. Subsequently, one or agroup (i.e., a predetermined number of) non-landmark data point(s) withlongest first landmark distance(s) in comparison with other firstlandmark distances of other non-landmark data points may be identified.The non-landmark data point(s) associated with the longest firstlandmark distance as a first landmark point may be added to the initialset of landmarks to generate an expanded set of landmark points.

The resolution module 318 may generate a cover of R based on theresolution received from the user (e.g., filter(s), intervals, andoverlap—see discussion regarding FIG. 7 for example). The cover of R maybe a finite collection of open sets (in the metric of R) such that everypoint in R lies in at least one of these sets.

Having computed, for each landmark point, which “cover tags” it isassigned to, for each cover element, C_(d), the points may beconstructed, whose tags included, as set S(d). This may mean that everylandmark point s is in S(d) for some d, but some landmark points maybelong to more than one such set. In some embodiments, there is,however, no requirement that each S(d) is non-empty, and it isfrequently the case that some of these sets are empty. In thenon-parallelized version of some embodiments, each landmark point x isprocessed in turn, and x is inserted into a hash-bucket for each j inref_tags(t) (that is, this may be how S(d) sets are computed).

The analysis module 320 may cluster each landmark S(d) based on themetric, filter, and the space S. In some embodiments, a dynamicsingle-linkage clustering algorithm may be used to partition S(d).

The visualization engine 322 may identify nodes which are associatedwith a subset of the partition elements of all of the landmark S(d) forgenerating a visualization. Of the sets of points that are used, twonodes intersect provided that the associated node sets have a non-emptyintersection.

The visualization engine 322 may join clusters to identify edges (e.g.,connecting lines between nodes). Once the nodes are constructed, theintersections (e.g., edges) may be computed “all at once,” by computing,for each point, the set of node sets (not ref_tags, this time). That is,for each landmark s in S, node_id_set(s) may be computed, which is anint[ ]. In some embodiments, if the cover is well behaved, then thisoperation is linear in the size of the set S, and we then iterate overeach pair in node_id_set(s). There may be an edge between two node_id'sif they both belong to the same node_id_set( ) value, and the number oflandmark points in the intersection is precisely the number of differentnode_id sets in which that pair is seen. This means that, except for theclustering step (which is often quadratic in the size of the sets S(d),but whose size may be controlled by the choice of cover), all of theother steps in the graph construction algorithm may be linear in thesize of S, and may be computed quite efficiently.

The visualization engine 322 may generate the visualization ofinterconnected nodes.

The landmark process may be used at other stages in topologicalanalysis. For example, nodes may be determined based on complex datausing topological data analysis as described herein. The nodes may alsobe landmarked and a visualization may be generated that includes thenodes of landmark points. This subset of nodes may have in a mannersimilar to the larger set of all nodes.

In one example, as discussed regarding FIG. 8, the input module 314 mayreceive data (e.g., data S). In one example, a user identifies a datastructure and then identifies ID and data fields. Data S may be based onthe information within the ID and data fields. The input module 314 maygenerate reference space R. In one example, reference space R may be awell-known metric space (e.g., such as the real line). The analysismodule 320 may generate a map ref( ) from S into R. The map ref( ) fromS into R may be called the “reference map.”

The resolution module 318 may generate a cover of R based on theresolution received from the user (e.g., filter(s), intervals, andoverlap—see discussion regarding FIG. 7 for example). The cover of R maybe a finite collection of open sets (in the metric of R) such that everypoint in R lies in at least one of these sets.

The analysis module 320 may cluster each data point S(d) based on themetric, filter, and the space S. In some embodiments, a dynamicsingle-linkage clustering algorithm may be used to partition S(d).

The visualization engine 322 may identify nodes which are associatedwith a subset of the partition elements of all of the data points S(d)for generating a visualization. Of the sets of points that are used, twonodes intersect provided that the associated node sets have a non-emptyintersection.

The nodes may be landmarked. For example, an initial set of nodes may beidentified as landmark nodes. For each non-landmark node, first datapoint distances between a respective non-landmark node and each landmarknode of the initial set of landmarks may be calculated, a first shortestdata point distance from among the first data point distances betweenthe respective non-landmark node and each landmark node of the initialset of landmarks may be identified, and the first shortest data pointdistance as a first landmark distance for the respective non-landmarknode may be stored. Subsequently, one or a group (i.e., a predeterminednumber of) non-landmark data point(s) with longest first landmarkdistance(s) in comparison with other first landmark distances of othernon-landmark nodes may be identified. The non-landmark node(s)associated with the longest first landmark distance as a first landmarknode may be added to the initial set of landmarks to generate anexpanded set of landmark nodes.

The visualization engine 322 may join clusters to identify edges (e.g.,connecting lines between nodes). Once the nodes are constructed, theintersections (e.g., edges) may be computed “all at once,” by computing,for each point, the set of node sets. The visualization engine 322 maygenerate the visualization of interconnected nodes.

FIG. 24A shows an example wherein data in X does not fit into localmemory 2402 (e.g., Random Access Memory (RAM)) and is, therefore, readoff of long term storage 2404. In this example, landmark set 2408represents storage of the set of all landmark points and data point sets2406 a, 2406 b, and 2406 c represent three different portions of thedata space X (e.g., each of data points sets 2406 a, 2406 b, and 2406 ccontaining different data). Accordingly, in this example, landmark set2408 and only a first set 2406 a of data space X can fit in local memory2402.

In the example discussed herein, landmark set 2408 could represent anamount of data points on the order of about 5,000 points and data pointsets 2406 a, 2406 b, and 2406 c could represent an amount of data pointson the order of about 100 million+ data points. Thus, once data pointset 2406 a has been compared to landmark set 2408 to determine thedistance calculations, data point set 2406 a must be removed from localmemory 2402 to make room for data point set 2406 b. After removal ofdata point set 2406 a from local memory 2402, data point set 2406 b isread off disk 2404 and loaded into local memory 2402. Accordingly, oncedata point set 2406 b has been compared to landmark set 2408 todetermine those distance calculations, data point set 2406 b is removedfrom local memory 2402 and data point set 2406 c is read off disk 2404and loaded into local memory 2402. The process of reading this much dataoff of disk 2404 creates significant latency.

Since the number of landmark points does not change until after a newlandmark has been determined and added (i.e., after each iteration), thenumber of landmark points is effectively limited for each round ofdistance calculations and, thus, PROCESS_x_AND_l( ) may only depend on x(e.g., data point sets 2406 a, 2406 b, and 2406 c). Therefore,PROCESS_x_AND_l( ) can be called in any order on x and the set L values(Landmark point values), provided that the process is being called onall landmark (landmark set 2408) and non-landmark point pairs (e.g.,data point sets 2406 a, 2406 b, and 2406 c). As a result, the firstprocess described above may be reordered to process all landmark points(e.g., landmark set 2408) for each x (e.g., data point sets 2406 a, 2406b, and 2406 c) instead of all points in X for each landmark point L.Accordingly, FIG. 24B shows an example wherein data point sets 2406 a,2406 b, and 2406 c are stored in local memory 2402 instead of landmarkset 2408 in accordance with various embodiments. Thus, instead ofrepeatedly reading the large amount of data associated with data pointsets 2406 a, 2406 b, and 2406 c off of disk 2404, the comparatively muchsmaller amount of data associated with landmark set 2408 is read offdisk 2404. Thus, the first portion of a method for generating a set oflandmark points may be reordered (STEP1A) and implemented as follows:

for x = 0, x < |X|, x++ do for l = 0, l < |R|, l++ doPROCESS_x_AND_l(x,l) done

Since PROCESS_x_AND_l( ) only depends on x, a current state or snapshotof the set of landmarks can be stored in local memory 2402 andPROCESS_x_AND_l( ) can be altered to use that state when performing anext iteration of distance calculations. Accordingly, if that state fitsinto local memory 2402 along with, for example, J rows of X, and thereordered first portion (STEP1A) of the method for generating the set oflandmarks and be run with only |X|/J page faults. For example, since thenumber of landmark points L is generally much smaller than the set ofdata points |X|, the number of page faults when scanning X with L inlocal memory 2402 is approximately the same as the number of page faultswhen scanning X without L. For example, if M is a number of rows of Xwhich can be simultaneously stored in local memory 2402, then the numberof page faults associated with the first potion of the method (STEP1)before reordering is approximately |L|*|X|/M and the number of pagefaults associated with the first portion of the method after reordering(STEP1A) is approximately |X|/(M−|L|).

Further, given T threads, the data points of X can be split into stripessuch that at least T of these stripes can fit into local memory 2402along with L. Accordingly, each thread of T can independently process astripe, such that there are ‘T versions’ of STEP1A concurrentlyoperating. As the x values are partitioned, contention is minimal and wesee in practice speedups of a factor of T. Concurrent operations do notalways finish precisely at the same time, thus, in one example, eachthread may include spin-locks to acquire new a stripe in order. This canalso enable the stripes to be fairly small and kept roughly together asX is iterated over.

The max-min landmark selection process for T threads is somewhatdifferent, but it can be understood as a FIND_NEXT_L(1) followed by aSTEP1 with only one landmark (which is equivalent to STEP1A, in thiscase) instead of a STEP2. This means that the end of each STEP1A threadcan be synchronized to then run a FIND_NEXT_L( ) and then partition Xinto stripes and run the STEP1A piece in parallel. As FIND_NEXT_L( )iterates over two (or more) arrays (e.g., one of booleans and another ofdoubles or floats), it may have paging issues only for truly giganticspaces or machines with small amounts of memory.

In various embodiments, instead of determining a single new landmarkpoint for each iteration of the aforementioned method of generating aset of landmark points, multiple landmark points can be chosen at atime. FIGS. 25A-25C show a process for generating a set of landmarkpoints wherein multiple landmark points are selected for each iterationof distance calculations described above. In at least one embodiment,for each iteration of distance calculations, the top “n” data pointsassociated with the longest distances to a nearest landmark point couldbe selected. The number “n” could vary, such as with the size of dataspace X, or it could be fixed to select a top predetermined number(e.g., 5) of the most distant data points, for example, from arespective landmark point for each iteration of distance calculations.

FIG. 25A shows subset 2102 with distances shown for points P₁, P₂, P₃,and P₄ to their respective closest random landmark (R₁, R₂, R₃, R₄). Inthis example, distance d₁ between P₁ and R₁ is 3, distance d₂ between P₁and R₂ is 5, distance d₃ between P₃ and R₃ is 7, and distance d₄ betweenP₄ and R₂ is 4 and these distances are shown in table 2550 of FIG. 25B.FIG. 25B shows example shortest distances from each non-landmark pointto each landmark point.

In this example, points P₂, P₃, and P₄ are in a top “n” data pointsbeing selected for this iteration based on each of their correspondingdistances to their nearest landmark point. For example, among an “n”number of landmark points being selected for this particular iteration,the distance d₁=3, between P₁ and R₁, is too short relative to otherdata points in subset 2102 and may not, therefore, be chosen forinclusion in the set of landmark points. Points P₂, P₃, and P₄, however,are chosen for inclusion in the set of landmark points with randomlandmark points (R₁, R₂, R₃, R₄).

Accordingly, FIG. 25C shows points P₂, P₃, and P₄ as landmarks L₁, L₂,and L₃ in this example. As can be seen in FIG. 25C, L₁ and L₂ are closetogether since they were selected without taking their relativedistances to each other into consideration and at least one of themwould not have been chose as a landmark point if only one landmark werechosen at a time. However, this process may work as an approximation andthe landmark points may not necessarily need to be perfectly spaced whenthe collection of data points is large. One way to potentially avoidchoosing landmarks that are too close to each other is by firstselecting a single landmark in a first iteration, a few such as 5landmarks in a second iteration, a single landmark again in a thirditeration, and so on. However, even if a few landmark points end upbeing close to each other, when taken into account with all otherlandmarks, the space can still be effectively approximated.

In some embodiments, more than one landmark point can be selected at atime by executing STEP1A on all landmarks at once, further resulting infewer iterations over X. In this example, identifying multiple MMlandmarks at a time may be accomplished by noticing that values ofdClosest[ ] decrease as more landmark points are added. Thus, the valuesof dClosest[x] may only stay the same or go down as more data points areadded to the set of landmark points. The landmark at l is, thus, the xin dClosest[ ] at step l−1 which has the largest value. As a result, ifx is the MM landmark at l, an obvious candidate for the MM landmark atl+1 may be the x′ which has the second largest value in dClosest[ ] atl−1. In one example, if dClosest[x′] does not decrease, x′ will be thelandmark point chosen at l+1 using any of the aforementioned processesfor selecting landmark points. In other words, if x′ is further from xthan from the closest of the previous l−1 landmark points, then it maybe the l+1th landmark. This pruning can be extended by remembering somefixed number K of largest indices and values for dClosest[ ], and thenpruning these by various heuristic processes. For instance, STEP2 fromabove can be altered as the following:

double dist = distance(x, L[l]); if (dist < dClosest[x]) { dClosest[x] =dist; insertKLargest(x, dist);

In this example, insertKLargest( ) maintains a data structure whichrecalls the K-largest pairs (x,distance). We can then iterate over the Kpairs, largest first, to recompute the dClosest[ ] values by adding thepoint with associated with the largest distance to the set of landmarkpoints. Any values which remain larger than other dClosest[ ] values canbe considered reliable and values which remain as the process continuesto add additional points to the set of landmark points as the values ofdClosest[ ] are adjusted along the way are themselves the landmarkpoints this process is searching for. This process might fail to findany additional landmark points, however, as all the K-largest pairsmight be part of a cluster eliminated by a newest landmark in theprocess. In practice, however, this process generally results inadditional landmark points, and can reduce the number of iterations overX by the average number of landmarks generated.

In one example, the distance calculations can use only the smallestvalues of the distances for a given data point x, such that the numbervaries depending on the K in STEP2. The following method (STEP2A) isequivalent, and for certain spaces, can be more efficient:

double dist = distanceUpToLimit(x, L[l], KLargest(x)); if (dist <dClosest[x]) { dClosest[x] = dist; insertKLargest(x, dist);

In this example, distanceUpToLimit( ) will quit calculating the distancewhen it is known that the distance may be “too large to be interesting.”Since many points can be considered relatively far away, and spaces ofdimensionality in the millions are not uncommon (and those in thethousands and tens of thousands are routine), this can lead tosignificant performance improvements.

In another example, the distances within a stripe can be computed in asingle pass where the computation for each pairwise distances areinterleaved, rather than computed serially. Such a process can beutilized at a low level and useful when using a smaller number ofthreads and metrics for which the distances can make use of specializedvectorization hardware. This approach has the potential to deliverimproved performance, as it eliminates a lot of the redundantcomputations introduced when each distance in a stripe is computedserially. Testing, however, indicates that under load, the performanceadvantage associated with interleaving is marginalized, as threads spendan increasing amount of time waiting for the memory subsystem torespond.

Accordingly, interleaving, in effect, may do a kind of loop unrolling atthe lowest level of the metric calculation. For example:

double 12(double *in0, double *in1, int len) { double accum = 0; for(int i = 0; i < len; i++) {  accum += (in0[i] − in1[i]) * (in0[i] −in1[i]); } return sqrt(accum);  }  void interleaved_l2(double *x0,double *x1, double *x2, double *x3, double *y, int len, double *accum) {double accum0 = 0, accum1 = 0, accum2 = 0, accum3 = 0; for (i = 0; i <len; i++) { double yval = y[i]; accum0 += (x0[i] − yval) * (x0[i] −yval); accum1 += (x1[i] − yval) * (x1[i] − yval); accum2 += (x2[i] −yval) * (x2[i] − yval); accum3 += (x3[i] − yval) * (x3[i] − yval); }accum[0] = sqrt(accum0); accum[1] = sqrt(accum1); accum[2] =sqrt(accum2); accum[3] = sqrt(accum3);

In this example, “yval” does not need to be reloaded and the process wasable to avoid doing three of the four loop checks. The larger “len” isthe more this will matter. Thus, in this example, the distances may allbe computed in a single pass where the computation of each pairwisedistance was interleaved, rather than computed serially. Accordingly,this approach eliminates some redundant computations introduced wheneach distance in a stripe is computed serially, thereby, increasingcomputational efficiency.

The above-described functions and components can be comprised ofinstructions that are stored on a storage medium (e.g., a computerreadable storage medium). The instructions can be retrieved and executedby a processor. Some examples of instructions are software, programcode, and firmware. Some examples of storage medium are memory devices,tape, disks, integrated circuits, and servers. The instructions areoperational when executed by the processor (e.g., a data processingdevice) to direct the processor to operate in accord with embodiments ofthe present invention. Those skilled in the art are familiar withinstructions, processor(s), and storage medium.

The present invention has been described above with reference toexemplary embodiments. It will be apparent to those skilled in the artthat various modifications may be made and other embodiments can be usedwithout departing from the broader scope of the invention. Therefore,these and other variations upon the exemplary embodiments are intendedto be covered by the present invention.

What is claimed is:
 1. A method comprising: receiving data points;selecting a first subset of the data points to generate an initial setof landmarks, each data point of the first subset defining a landmarkpoint; for each non-landmark data point, calculating first data pointdistances between a respective non-landmark data point and each landmarkpoint of the initial set of landmarks; identifying a first shortest datapoint distance from among the first data point distances between therespective non-landmark data point and each landmark point of theinitial set of landmarks; and storing the first shortest data pointdistance as a first landmark distance for the respective non-landmarkdata point; identifying a non-landmark data point with a longest firstlandmark distance in comparison with other first landmark distances ofother non-landmark data points; and adding the non-landmark data pointassociated with the longest first landmark distance as a first landmarkpoint to the initial set of landmarks to generate an expanded set oflandmark points.
 2. The method of claim 1 further comprising: storing aset of longest landmark distances for the non-landmark data points ofthe expanded set of landmark points; calculating second data pointdistances between the non-landmark data points associated with the setof longest landmark distances and each landmark point of the expandedset of landmark points, the expanded set of landmark points includingthe first landmark point; and adding the non-landmark data pointassociated with the longest second landmark distance to the expanded setof landmark points, thereby categorizing the non-landmark data point asa new landmark point.
 3. The method of claim 2 further comprising:determining if the expanded set of landmark points includes apredetermined number of landmark points; if the expanded set of landmarkpoints includes less than the predetermined number of landmark points:storing another set of longest second landmark distances for thenon-landmark data points of the expanded set of landmark points;calculating another data point distances between the non-landmark datapoints associated with the set of longest second landmark distances andeach landmark point of the first and second sets of landmark points, thefirst set and second set of landmark points including the first andsecond landmark point; and adding the non-landmark data point associatedwith the longest second landmark distance as a another landmark point tothe set of landmark points; and if the expanded set of landmark pointsincludes less than the predetermined number of landmark points, generatea visualization of nodes and edges using the expanded set of landmarkpoints instead of the received data points.
 4. The method of claim 1,wherein the expanded set of landmark points is an approximation of thereceived data points.
 5. The method of claim 1, further comprising:before the first subset of the data points are selected to generate aninitial set of landmarks: accessing a data structure containinginformation; generating a mathematical reference space; and mapping asimilarity space using the information from the data structure into themathematical reference space to generate the data points, wherein eachdata point is defined by the mapping into the reference space.
 6. Themethod of claim 5, further comprising: generating a cover for thelandmark points of the expanded set of landmark points in themathematical reference space based on a resolution; clustering thelandmark points of the expanded set of landmark points into subsetsusing a metric to generate subsets of the landmark points of theexpanded set of landmark points to determine each individual node of aplurality of nodes, each of the nodes of the plurality of nodescomprising members representative of at least one subset of the landmarkpoints of the expanded set of landmark points; and generating aninteractive visualization comprising nodes and a plurality of edgeswherein each of the edges of the plurality of edges connects nodes withshared members.
 7. The method of claim 5, further comprising: generatinga cover for the data points in the mathematical reference space based ona resolution; clustering the data points into subsets using a metric togenerate subsets of the data points to determine each individual node ofa plurality of nodes, each of the nodes of the plurality of nodescomprising members representative of at least one subset of the datapoints, each node defining a data point; and generating an interactivevisualization comprising nodes of landmark points of the expanded set oflandmark points and a plurality of edges wherein each of the edges ofthe plurality of edges connects nodes with shared members.
 8. The methodof claim 1 wherein selection of the first subset of the data point israndom.
 9. The method of claim 1, wherein receiving data pointscomprises storing the data points in a memory system and the initial setof landmarks is stored in a non-transitory storage system.
 10. Themethod of claim 1, wherein identifying the non-landmark data point withthe longest first landmark distance in comparison with other firstlandmark distances of other non-landmark data points comprisesidentifying a predetermined number of non-landmark data points, each ofthe predetermined number having longer first landmark distances incomparison with the other first landmark distances of the othernon-landmark data points.
 11. The method of claim 10, wherein adding thenon-landmark data point associated with the longest first landmarkdistance as a first landmark point to the initial set of landmarkscomprises adding the predetermined number of non-landmark data pointsassociated with the longer first landmark distances to the initial setof landmarks.
 12. A non-transitory computer readable medium comprisinginstructions executable by a processor to perform a method, the methodcomprising: receiving data points; selecting a first subset of the datapoints to generate an initial set of landmarks, each data point of thefirst subset defining a landmark point; for each non-landmark datapoint, calculating first data point distances between a respectivenon-landmark data point and each landmark point of the initial set oflandmarks; identifying a first shortest data point distance from amongthe first data point distances between the respective non-landmark datapoint and each landmark point of the initial set of landmarks; andstoring the first shortest data point distance as a first landmarkdistance for the respective non-landmark data point; identifying anon-landmark data point with a longest first landmark distance incomparison with other first landmark distances of other non-landmarkdata points; and adding the non-landmark data point associated with thelongest first landmark distance as a first landmark point to the initialset of landmarks to generate an expanded set of landmark points.
 13. Thenon-transitory computer readable medium of claim 12, the method furthercomprising: storing a set of longest landmark distances for thenon-landmark data points of the expanded set of landmark points;calculating second data point distances between the non-landmark datapoints associated with the set of longest landmark distances and eachlandmark point of the expanded set of landmark points, the expanded setof landmark points including the first landmark point; and adding thenon-landmark data point associated with the longest second landmarkdistance to the expanded set of landmark points, thereby categorizingthe non-landmark data point as a new landmark point.
 14. Thenon-transitory computer readable medium of claim 13, the method furthercomprising: determining if the expanded set of landmark points includesa predetermined number of landmark points; if the expanded set oflandmark points includes less than the predetermined number of landmarkpoints: storing another set of longest second landmark distances for thenon-landmark data points of the expanded set of landmark points;calculating another data point distances between the non-landmark datapoints associated with the set of longest second landmark distances andeach landmark point of the first and second sets of landmark points, thefirst set and second set of landmark points including the first andsecond landmark point; and adding the non-landmark data point associatedwith the longest second landmark distance as a another landmark point tothe set of landmark points; and if the expanded set of landmark pointsincludes less than the predetermined number of landmark points, generatea visualization of nodes and edges using the expanded set of landmarkpoints instead of the received data points.
 15. The non-transitorycomputer readable medium of claim 12, the method further comprising:before the first subset of the data points are selected to generate aninitial set of landmarks: accessing a data structure containinginformation; generating a mathematical reference space; and mapping asimilarity space using the information from the data structure into themathematical reference space to generate the data points, wherein eachdata point is defined by the mapping into the reference space.
 16. Thenon-transitory computer readable medium of claim 15, the method furthercomprising: generating a cover for the landmark points of the expandedset of landmark points in the mathematical reference space based on aresolution; clustering the landmark points of the expanded set oflandmark points into subsets using a metric to generate subsets of thelandmark points of the expanded set of landmark points to determine eachindividual node of a plurality of nodes, each of the nodes of theplurality of nodes comprising members representative of at least onesubset of the landmark points of the expanded set of landmark points;and generating an interactive visualization comprising nodes and aplurality of edges wherein each of the edges of the plurality of edgesconnects nodes with shared members.
 17. The non-transitory computerreadable medium of claim 15, the method further comprising: generating acover for the data points in the mathematical reference space based on aresolution; clustering the data points into subsets using a metric togenerate subsets of the data points to determine each individual node ofa plurality of nodes, each of the nodes of the plurality of nodescomprising members representative of at least one subset of the datapoints, each node defining a data point; and generating an interactivevisualization comprising nodes of landmark points of the expanded set oflandmark points and a plurality of edges wherein each of the edges ofthe plurality of edges connects nodes with shared members.
 18. Thenon-transitory computer readable medium of claim 12, wherein identifyingthe non-landmark data point with the longest first landmark distance incomparison with other first landmark distances of other non-landmarkdata points comprises identifying a predetermined number of non-landmarkdata points, each of the predetermined number having longer firstlandmark distances in comparison with the other first landmark distancesof the other non-landmark data points.
 19. The non-transitory computerreadable medium of claim 18, wherein adding the non-landmark data pointassociated with the longest first landmark distance as a first landmarkpoint to the initial set of landmarks comprises adding the predeterminednumber of non-landmark data points associated with the longer firstlandmark distances to the initial set of landmarks.
 20. A systemcomprising: an input module configured to receive data points; a randomlandmark selection module configured to select a first subset of thedata points to generate an initial set of landmarks, each data point ofthe first subset defining a landmark point; a distance calculationmodule configured to, for each non-landmark data point, calculate firstdata point distances between a respective non-landmark data point andeach landmark point of the initial set of landmarks; identify a firstshortest data point distance from among the first data point distancesbetween the respective non-landmark data point and each landmark pointof the initial set of landmarks; and store the first shortest data pointdistance as a first landmark distance for the respective non-landmarkdata point; a landmark distance comparison module configured to identifya non-landmark data point with a longest first landmark distance incomparison with other first landmark distances of other non-landmarkdata points; and a landmark assignment module configured to add thenon-landmark data point associated with the longest first landmarkdistance as a first landmark point to the initial set of landmarks togenerate an expanded set of landmark points.